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</script></head><body><div id="package-header"><ul class="links" id="page-menu"><li><a href="src/Math-Algebra-Group-PermutationGroup.html">Source</a></li><li><a href="index.html">Contents</a></li><li><a href="doc-index.html">Index</a></li></ul><p class="caption">HaskellForMaths-0.4.5: Combinatorics, group theory, commutative algebra, non-commutative algebra</p></div><div id="content"><div id="module-header"><table class="info"><tr><th>Safe Haskell</th><td>None</td></tr></table><p class="caption">Math.Algebra.Group.PermutationGroup</p></div><div id="description"><p class="caption">Description</p><div class="doc"><p>A module for doing arithmetic in permutation groups.
</p><p>Group elements are represented as permutations of underlying sets, and are entered and displayed
using a Haskell-friendly version of cycle notation. For example, the permutation (1 2 3)(4 5)
would be entered as <code>p [[1,2,3],[4,5]]</code>, and displayed as [[1,2,3],[4,5]]. Permutations can be defined
over arbitrary underlying sets (types), not just the integers.
</p><p>If <code>g</code> and <code>h</code> are group elements, then the expressions <code>g*h</code> and <code>g^-1</code> calculate product and inverse respectively.
</p></div></div><div id="synopsis"><p id="control.syn" class="caption expander" onclick="toggleSection('syn')">Synopsis</p><ul id="section.syn" class="hide" onclick="toggleSection('syn')"><li class="src short"><a href="#v:rotateL">rotateL</a> :: [a] -> [a]</li><li class="src short"><span class="keyword">newtype</span> <a href="#t:Permutation">Permutation</a> a = <a href="#v:P">P</a> (<a href="/usr/share/doc/ghc-doc/html/libraries/containers-0.5.0.0/Data-Map-Lazy.html#t:Map">Map</a> a a)</li><li class="src short"><a href="#v:p">p</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [[a]] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:fromPairs">fromPairs</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [(a, a)] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:fromPairs-39-">fromPairs'</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [(a, a)] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:toPairs">toPairs</a> :: <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [(a, a)]</li><li class="src short"><a href="#v:fromList">fromList</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:supp">supp</a> :: <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [a]</li><li class="src short"><a href="#v:.-94-">(.^)</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> a</li><li class="src short"><a href="#v:-45--94-">(-^)</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [a]</li><li class="src short"><a href="#v:fromCycles">fromCycles</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [[a]] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:toCycles">toCycles</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> t -> [[t]]</li><li class="src short"><a href="#v:cycleOf">cycleOf</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> a -> [a]</li><li class="src short"><a href="#v:parity">parity</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a></li><li class="src short"><a href="#v:sign">sign</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a1) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a1 -> a</li><li class="src short"><a href="#v:orderElt">orderElt</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a></li><li class="src short"><a href="#v:-126--94-">(~^)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:comm">comm</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="Math-Core-Utils.html#t:HasInverses">HasInverses</a> a) => a -> a -> a</li><li class="src short"><a href="#v:closureS">closureS</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> [a -> a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/containers-0.5.0.0/Data-Set.html#t:Set">Set</a> a</li><li class="src short"><a href="#v:closure">closure</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> [a -> a] -> [a]</li><li class="src short"><a href="#v:orbit">orbit</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => (a -> t -> a) -> a -> [t] -> [a]</li><li class="src short"><a href="#v:.-94--94-">(.^^)</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a]</li><li class="src short"><a href="#v:orbitP">orbitP</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> a -> [a]</li><li class="src short"><a href="#v:orbitV">orbitV</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> a -> [a]</li><li class="src short"><a href="#v:-45--94--94-">(-^^)</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [[a]]</li><li class="src short"><a href="#v:orbitB">orbitB</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a] -> [[a]]</li><li class="src short"><a href="#v:orbitE">orbitE</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a] -> [[a]]</li><li class="src short"><a href="#v:action">action</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> (a -> a) -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:orbits">orbits</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> t] -> [[t]]</li><li class="src short"><a href="#v:_C">_C</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:_D">_D</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:_D2">_D2</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:_S">_S</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:_A">_A</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:dp">dp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> b) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> b] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Either.html#t:Either">Either</a> a b)]</li><li class="src short"><a href="#v:wr">wr</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> t] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> (t, a)]</li><li class="src short"><a href="#v:toSn">toSn</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Enum">Enum</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a1) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a1]</li><li class="src short"><a href="#v:fromDigits">fromDigits</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:fromDigits-39-">fromDigits'</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a => [a] -> a</li><li class="src short"><a href="#v:fromBinary">fromBinary</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:fromBinary-39-">fromBinary'</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a => [a] -> a</li><li class="src short"><a href="#v:elts">elts</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> [a]</li><li class="src short"><a href="#v:eltsS">eltsS</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/containers-0.5.0.0/Data-Set.html#t:Set">Set</a> a</li><li class="src short"><a href="#v:order">order</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a></li><li class="src short"><a href="#v:isMember">isMember</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> a -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:minsupp">minsupp</a> :: <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> c -> c</li><li class="src short"><a href="#v:orderTGS">orderTGS</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> a1</li><li class="src short"><a href="#v:eltsTGS">eltsTGS</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:tgsFromSgs">tgsFromSgs</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:orderSGS">orderSGS</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integer">Integer</a></li><li class="src short"><a href="#v:gens">gens</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> [a]</li><li class="src short"><a href="#v:-126--94--94-">(~^^)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:conjClass">conjClass</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:conjClassReps">conjClassReps</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [(<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>)]</li><li class="src short"><a href="#v:reduceGens">reduceGens</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> [a]</li><li class="src short"><a href="#v:isSubgp">isSubgp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> [a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:subgps">subgps</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [[<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]]</li><li class="src short"><a href="#v:isMinimal">isMinimal</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:centralizer">centralizer</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> t, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t) => [t] -> [t] -> [t]</li><li class="src short"><a href="#v:centre">centre</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> t, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t) => [t] -> [t]</li><li class="src short"><a href="#v:normalizer">normalizer</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:stabilizer">stabilizer</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:ptStab">ptStab</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:setStab">setStab</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:normalClosure">normalClosure</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:commutatorGp">commutatorGp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:derivedSubgp">derivedSubgp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:-45--42--45-">(-*-)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> b, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> b) => [b] -> [b] -> [b]</li><li class="src short"><a href="#v:-45--42-">(-*)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> a -> [a]</li><li class="src short"><a href="#v:-42--45-">(*-)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => a -> [a] -> [a]</li><li class="src short"><a href="#v:isNormal">isNormal</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:normalSubgps">normalSubgps</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [[<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]]</li><li class="src short"><a href="#v:isSimple">isSimple</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a></li><li class="src short"><a href="#v:cosets">cosets</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> t, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t) => [t] -> [t] -> [[t]]</li><li class="src short"><a href="#v:quotientGp">quotientGp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>]</li><li class="src short"><a href="#v:-47--47-">(//)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>]</li><li class="src short"><a href="#v:-126--126--94-">(~~^)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]</li><li class="src short"><a href="#v:conjugateSubgps">conjugateSubgps</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [[<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]]</li><li class="src short"><a href="#v:subgpAction">subgpAction</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Enum">Enum</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a1]</li><li class="src short"><a href="#v:rrpr">rrpr</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:rrpr-39-">rrpr'</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a</li><li class="src short"><a href="#v:permutationMatrix">permutationMatrix</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> t, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [[t]]</li></ul></div><div id="interface"><h1>Documentation</h1><div class="top"><p class="src"><a name="v:rotateL" class="def">rotateL</a> :: [a] -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#rotateL" class="link">Source</a></p></div><div class="top"><p class="src"><span class="keyword">newtype</span> <a name="t:Permutation" class="def">Permutation</a> a <a href="src/Math-Algebra-Group-PermutationGroup.html#Permutation" class="link">Source</a></p><div class="doc"><p>A type for permutations, considered as functions or actions which can be performed on an underlying set.
</p></div><div class="subs constructors"><p class="caption">Constructors</p><table><tr><td class="src"><a name="v:P" class="def">P</a> (<a href="/usr/share/doc/ghc-doc/html/libraries/containers-0.5.0.0/Data-Map-Lazy.html#t:Map">Map</a> a a)</td><td class="doc empty"> </td></tr></table></div><div class="subs instances"><p id="control.i:Permutation" class="caption collapser" onclick="toggleSection('i:Permutation')">Instances</p><div id="section.i:Permutation" class="show"><table><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Eq.html#t:Eq">Eq</a> k, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> k) => <a href="Math-Algebras-Structures.html#t:HopfAlgebra">HopfAlgebra</a> k (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Eq.html#t:Eq">Eq</a> k, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> k) => <a href="Math-Algebras-Structures.html#t:Bialgebra">Bialgebra</a> k (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Eq.html#t:Eq">Eq</a> k, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> k) => <a href="Math-Algebras-Structures.html#t:Coalgebra">Coalgebra</a> k (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Eq.html#t:Eq">Eq</a> k, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> k) => <a href="Math-Algebras-Structures.html#t:Algebra">Algebra</a> k (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Eq.html#t:Eq">Eq</a> k, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> k) => <a href="Math-Algebras-Structures.html#t:Module">Module</a> k (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>) <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a></td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Eq.html#t:Eq">Eq</a> k, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> k) => <a href="Math-Algebras-Structures.html#t:Module">Module</a> k (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>) [<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>]</td><td class="doc empty"> </td></tr><tr><td class="src"><a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Eq.html#t:Eq">Eq</a> a => <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Eq.html#t:Eq">Eq</a> (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a)</td><td class="doc"><p>The Num instance is what enables us to write <code>g*h</code> for the product of group elements and <code>1</code> for the group identity.
Unfortunately we can't of course give sensible definitions for the other functions declared in the Num typeclass.
</p></td></tr><tr><td class="src"><a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a)</td><td class="doc empty"> </td></tr><tr><td class="src">(<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="Math-Core-Utils.html#t:HasInverses">HasInverses</a> (<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a)</td><td class="doc"><p>The HasInverses instance is what enables us to write <code>g^-1</code> for the inverse of a group element.
</p></td></tr><tr><td class="src"><a href="Math-Core-Utils.html#t:HasInverses">HasInverses</a> (<a href="Math-Algebras-GroupAlgebra.html#t:GroupAlgebra">GroupAlgebra</a> <a href="Math-Core-Field.html#t:Q">Q</a>)</td><td class="doc"><p>Note that the inverse of a group algebra element can only be efficiently calculated
if the group generated by the non-zero terms is very small (eg <100 elements).
</p></td></tr></table></div></div></div><div class="top"><p class="src"><a name="v:p" class="def">p</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [[a]] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#p" class="link">Source</a></p><div class="doc"><p>Construct a permutation from a list of cycles.
For example, <code>p [[1,2,3],[4,5]]</code> returns the permutation that sends 1 to 2, 2 to 3, 3 to 1, 4 to 5, 5 to 4.
</p></div></div><div class="top"><p class="src"><a name="v:fromPairs" class="def">fromPairs</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [(a, a)] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#fromPairs" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:fromPairs-39-" class="def">fromPairs'</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [(a, a)] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#fromPairs%27" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:toPairs" class="def">toPairs</a> :: <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [(a, a)]<a href="src/Math-Algebra-Group-PermutationGroup.html#toPairs" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:fromList" class="def">fromList</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#fromList" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:supp" class="def">supp</a> :: <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#supp" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:.-94-" class="def">(.^)</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> a<a href="src/Math-Algebra-Group-PermutationGroup.html#.%5E" class="link">Source</a></p><div class="doc"><p>x .^ g returns the image of a vertex or point x under the action of the permutation g.
For example, <code>1 .^ p [[1,2,3]]</code> returns 2.
The dot is meant to be a mnemonic for point or vertex.
</p></div></div><div class="top"><p class="src"><a name="v:-45--94-" class="def">(-^)</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#-%5E" class="link">Source</a></p><div class="doc"><p>b -^ g returns the image of an edge or block b under the action of the permutation g.
For example, <code>[1,2] -^ p [[1,4],[2,3]]</code> returns [3,4].
The dash is meant to be a mnemonic for edge or line or block.
</p></div></div><div class="top"><p class="src"><a name="v:fromCycles" class="def">fromCycles</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [[a]] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#fromCycles" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:toCycles" class="def">toCycles</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> t -> [[t]]<a href="src/Math-Algebra-Group-PermutationGroup.html#toCycles" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:cycleOf" class="def">cycleOf</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> a -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#cycleOf" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:parity" class="def">parity</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a><a href="src/Math-Algebra-Group-PermutationGroup.html#parity" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:sign" class="def">sign</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a1) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a1 -> a<a href="src/Math-Algebra-Group-PermutationGroup.html#sign" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:orderElt" class="def">orderElt</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a><a href="src/Math-Algebra-Group-PermutationGroup.html#orderElt" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:-126--94-" class="def">(~^)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#~%5E" class="link">Source</a></p><div class="doc"><p>g ~^ h returns the conjugate of g by h, that is, h^-1*g*h.
The tilde is meant to a mnemonic, because conjugacy is an equivalence relation.
</p></div></div><div class="top"><p class="src"><a name="v:comm" class="def">comm</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="Math-Core-Utils.html#t:HasInverses">HasInverses</a> a) => a -> a -> a<a href="src/Math-Algebra-Group-PermutationGroup.html#comm" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:closureS" class="def">closureS</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> [a -> a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/containers-0.5.0.0/Data-Set.html#t:Set">Set</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#closureS" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:closure" class="def">closure</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> [a -> a] -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#closure" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:orbit" class="def">orbit</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => (a -> t -> a) -> a -> [t] -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#orbit" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:.-94--94-" class="def">(.^^)</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#.%5E%5E" class="link">Source</a></p><div class="doc"><p>x .^^ gs returns the orbit of the point or vertex x under the action of the gs
</p></div></div><div class="top"><p class="src"><a name="v:orbitP" class="def">orbitP</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> a -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#orbitP" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:orbitV" class="def">orbitV</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> a -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#orbitV" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:-45--94--94-" class="def">(-^^)</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [[a]]<a href="src/Math-Algebra-Group-PermutationGroup.html#-%5E%5E" class="link">Source</a></p><div class="doc"><p>b -^^ gs returns the orbit of the block or edge b under the action of the gs
</p></div></div><div class="top"><p class="src"><a name="v:orbitB" class="def">orbitB</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a] -> [[a]]<a href="src/Math-Algebra-Group-PermutationGroup.html#orbitB" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:orbitE" class="def">orbitE</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a] -> [[a]]<a href="src/Math-Algebra-Group-PermutationGroup.html#orbitE" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:action" class="def">action</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [a] -> (a -> a) -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#action" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:orbits" class="def">orbits</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> t] -> [[t]]<a href="src/Math-Algebra-Group-PermutationGroup.html#orbits" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:_C" class="def">_C</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#_C" class="link">Source</a></p><div class="doc"><p>_C n returns generators for Cn, the cyclic group of order n
</p></div></div><div class="top"><p class="src"><a name="v:_D" class="def">_D</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#_D" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:_D2" class="def">_D2</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#_D2" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:_S" class="def">_S</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#_S" class="link">Source</a></p><div class="doc"><p>_S n returns generators for Sn, the symmetric group on [1..n]
</p></div></div><div class="top"><p class="src"><a name="v:_A" class="def">_A</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integral">Integral</a> a => a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#_A" class="link">Source</a></p><div class="doc"><p>_A n returns generators for An, the alternating group on [1..n]
</p></div></div><div class="top"><p class="src"><a name="v:dp" class="def">dp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> b) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> b] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Either.html#t:Either">Either</a> a b)]<a href="src/Math-Algebra-Group-PermutationGroup.html#dp" class="link">Source</a></p><div class="doc"><p>Given generators for groups H and K, acting on sets A and B respectively,
return generators for the direct product H*K, acting on the disjoint union A+B (= Either A B)
</p></div></div><div class="top"><p class="src"><a name="v:wr" class="def">wr</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> t] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> (t, a)]<a href="src/Math-Algebra-Group-PermutationGroup.html#wr" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:toSn" class="def">toSn</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Enum">Enum</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a1) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a1]<a href="src/Math-Algebra-Group-PermutationGroup.html#toSn" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:fromDigits" class="def">fromDigits</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#fromDigits" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:fromDigits-39-" class="def">fromDigits'</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a => [a] -> a<a href="src/Math-Algebra-Group-PermutationGroup.html#fromDigits%27" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:fromBinary" class="def">fromBinary</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#fromBinary" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:fromBinary-39-" class="def">fromBinary'</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a => [a] -> a<a href="src/Math-Algebra-Group-PermutationGroup.html#fromBinary%27" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:elts" class="def">elts</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#elts" class="link">Source</a></p><div class="doc"><p>Given generators for a group, return a (sorted) list of all elements of the group.
Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000)
</p></div></div><div class="top"><p class="src"><a name="v:eltsS" class="def">eltsS</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/containers-0.5.0.0/Data-Set.html#t:Set">Set</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#eltsS" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:order" class="def">order</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a><a href="src/Math-Algebra-Group-PermutationGroup.html#order" class="link">Source</a></p><div class="doc"><p>Given generators for a group, return the order of the group (the number of elements).
Implemented using a naive closure algorithm, so only suitable for small groups (|G| < 10000)
</p></div></div><div class="top"><p class="src"><a name="v:isMember" class="def">isMember</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> a -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a><a href="src/Math-Algebra-Group-PermutationGroup.html#isMember" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:minsupp" class="def">minsupp</a> :: <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> c -> c<a href="src/Math-Algebra-Group-PermutationGroup.html#minsupp" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:orderTGS" class="def">orderTGS</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> a1<a href="src/Math-Algebra-Group-PermutationGroup.html#orderTGS" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:eltsTGS" class="def">eltsTGS</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#eltsTGS" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:tgsFromSgs" class="def">tgsFromSgs</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#tgsFromSgs" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:orderSGS" class="def">orderSGS</a> :: <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Integer">Integer</a><a href="src/Math-Algebra-Group-PermutationGroup.html#orderSGS" class="link">Source</a></p><div class="doc"><p>Given a strong generating set, return the order of the group it generates
</p></div></div><div class="top"><p class="src"><a name="v:gens" class="def">gens</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#gens" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:-126--94--94-" class="def">(~^^)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#~%5E%5E" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:conjClass" class="def">conjClass</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#conjClass" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:conjClassReps" class="def">conjClassReps</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [(<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>)]<a href="src/Math-Algebra-Group-PermutationGroup.html#conjClassReps" class="link">Source</a></p><div class="doc"><p>conjClassReps gs returns conjugacy class representatives and sizes for the group generated by gs.
This implementation is only suitable for use with small groups (|G| < 10000).
</p></div></div><div class="top"><p class="src"><a name="v:reduceGens" class="def">reduceGens</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#reduceGens" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:isSubgp" class="def">isSubgp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> [a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a><a href="src/Math-Algebra-Group-PermutationGroup.html#isSubgp" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:subgps" class="def">subgps</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [[<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]]<a href="src/Math-Algebra-Group-PermutationGroup.html#subgps" class="link">Source</a></p><div class="doc"><p>Return the subgroups of a group. Only suitable for use on small groups (eg < 100 elts)
</p></div></div><div class="top"><p class="src"><a name="v:isMinimal" class="def">isMinimal</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a><a href="src/Math-Algebra-Group-PermutationGroup.html#isMinimal" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:centralizer" class="def">centralizer</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> t, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t) => [t] -> [t] -> [t]<a href="src/Math-Algebra-Group-PermutationGroup.html#centralizer" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:centre" class="def">centre</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> t, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t) => [t] -> [t]<a href="src/Math-Algebra-Group-PermutationGroup.html#centre" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:normalizer" class="def">normalizer</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#normalizer" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:stabilizer" class="def">stabilizer</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#stabilizer" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:ptStab" class="def">ptStab</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#ptStab" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:setStab" class="def">setStab</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#setStab" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:normalClosure" class="def">normalClosure</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#normalClosure" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:commutatorGp" class="def">commutatorGp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#commutatorGp" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:derivedSubgp" class="def">derivedSubgp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#derivedSubgp" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:-45--42--45-" class="def">(-*-)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> b, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> b) => [b] -> [b] -> [b]<a href="src/Math-Algebra-Group-PermutationGroup.html#-%2A-" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:-45--42-" class="def">(-*)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> a -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#-%2A" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:-42--45-" class="def">(*-)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => a -> [a] -> [a]<a href="src/Math-Algebra-Group-PermutationGroup.html#%2A-" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:isNormal" class="def">isNormal</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a><a href="src/Math-Algebra-Group-PermutationGroup.html#isNormal" class="link">Source</a></p><div class="doc"><p>isNormal gs ks returns True if <ks> is normal in <gs>.
Note, it is caller's responsibility to ensure that <ks> is a subgroup of <gs> (ie that each k is in <gs>).
</p></div></div><div class="top"><p class="src"><a name="v:normalSubgps" class="def">normalSubgps</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [[<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]]<a href="src/Math-Algebra-Group-PermutationGroup.html#normalSubgps" class="link">Source</a></p><div class="doc"><p>Return the normal subgroups of a group. Only suitable for use on small groups (eg < 100 elts)
</p></div></div><div class="top"><p class="src"><a name="v:isSimple" class="def">isSimple</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Bool.html#t:Bool">Bool</a><a href="src/Math-Algebra-Group-PermutationGroup.html#isSimple" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:cosets" class="def">cosets</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> t, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> t) => [t] -> [t] -> [[t]]<a href="src/Math-Algebra-Group-PermutationGroup.html#cosets" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:quotientGp" class="def">quotientGp</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>]<a href="src/Math-Algebra-Group-PermutationGroup.html#quotientGp" class="link">Source</a></p><div class="doc"><p>quotientGp gs ks returns <gs> / <ks>
</p></div></div><div class="top"><p class="src"><a name="v:-47--47-" class="def">(//)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Int.html#t:Int">Int</a>]<a href="src/Math-Algebra-Group-PermutationGroup.html#%2F%2F" class="link">Source</a></p><div class="doc"><p>Synonym for quotientGp
</p></div></div><div class="top"><p class="src"><a name="v:-126--126--94-" class="def">(~~^)</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]<a href="src/Math-Algebra-Group-PermutationGroup.html#~~%5E" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:conjugateSubgps" class="def">conjugateSubgps</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [[<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a]]<a href="src/Math-Algebra-Group-PermutationGroup.html#conjugateSubgps" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:subgpAction" class="def">subgpAction</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Enum">Enum</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a1, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Text-Show.html#t:Show">Show</a> a) => [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a] -> [<a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a1]<a href="src/Math-Algebra-Group-PermutationGroup.html#subgpAction" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:rrpr" class="def">rrpr</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#rrpr" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:rrpr-39-" class="def">rrpr'</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> a, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> a -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a<a href="src/Math-Algebra-Group-PermutationGroup.html#rrpr%27" class="link">Source</a></p></div><div class="top"><p class="src"><a name="v:permutationMatrix" class="def">permutationMatrix</a> :: (<a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Prelude.html#t:Num">Num</a> t, <a href="/usr/share/doc/ghc-doc/html/libraries/base-4.6.0.1/Data-Ord.html#t:Ord">Ord</a> a) => [a] -> <a href="Math-Algebra-Group-PermutationGroup.html#t:Permutation">Permutation</a> a -> [[t]]<a href="src/Math-Algebra-Group-PermutationGroup.html#permutationMatrix" class="link">Source</a></p></div></div></div><div id="footer"><p>Produced by <a href="http://www.haskell.org/haddock/">Haddock</a> version 2.13.2</p></div></body></html>
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