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<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap41.html">[Previous Chapter]</a> <a href="chap43.html">[Next Chapter]</a> </div>
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<p><a id="X80F808307A2D5AB8" name="X80F808307A2D5AB8"></a></p>
<div class="ChapSects"><a href="chap42.html#X80F808307A2D5AB8">42 <span class="Heading">Permutations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X80F07BE2811D4BAC">42.1 <span class="Heading">IsPerm (Filter)</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X7AA69C6686FC49EA">42.1-1 IsPerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X82069E437D2DF9AA">42.1-2 IsPermCollection</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X819628B083B3939B">42.1-3 PermutationsFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X7A21DE5779D21A6D">42.2 <span class="Heading">Comparison of Permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X7CEC03A9808E2E7C">42.2-1 \=</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X7BC944F57A04AFF2">42.2-2 DistancePerms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X83A917F67D45C7EA">42.2-3 SmallestGeneratorPerm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X82C255E2821C0721">42.3 <span class="Heading">Moved Points of Permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X84EF0A697F7A87DC">42.3-1 SmallestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X84AA603987C94AC0">42.3-2 LargestMovedPoint</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X85E61B9C7A6B0CCA">42.3-3 MovedPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X85E7B1E28430F49E">42.3-4 NrMovedPoints</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X79BE80267F4AA2B0">42.4 <span class="Heading">Sign and Cycle Structure</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X7BE5011B7C0DB704">42.4-1 SignPerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X7944D1447804A69A">42.4-2 CycleStructurePerm</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap42.html#X7B3194EC869D936D">42.5 <span class="Heading">Creating Permutations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X7A9DCFD986958C1E">42.5-1 ListPerm</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X78D611D17EA6E3BC">42.5-2 PermList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X8087DCC780B9656A">42.5-3 MappingPermListList</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap42.html#X7EF8388E7DA8E600">42.5-4 RestrictedPerm</a></span>
</div></div>
</div>
<h3>42 <span class="Heading">Permutations</span></h3>
<p><strong class="pkg">GAP</strong> offers a data type <em>permutation</em> to describe the elements of permutation groups.</p>
<p>The points on which permutations in <strong class="pkg">GAP</strong> act are the positive integers up to a certain architecture dependent limit, and the image of a point <span class="SimpleMath">i</span> under a permutation <span class="SimpleMath">p</span> is written <span class="SimpleMath">i^p</span>, which is expressed as <span class="SimpleMath">i</span><code class="code">^</code><span class="SimpleMath">p</span> in <strong class="pkg">GAP</strong>. (This action is also implemented by the function <code class="func">OnPoints</code> (<a href="chap41.html#X7FE417DD837987B4"><span class="RefLink">41.2-1</span></a>).) If <span class="SimpleMath">i</span><code class="code">^</code><span class="SimpleMath">p</span> is different from <span class="SimpleMath">i</span>, we say that <span class="SimpleMath">i</span> is <em>moved</em> by <span class="SimpleMath">p</span>, otherwise it is <em>fixed</em>. Permutations in <strong class="pkg">GAP</strong> are entered and displayed in cycle notation, such as <code class="code">(1,2,3)(4,5)</code>.</p>
<p>The preimage of the point <span class="SimpleMath">i</span> under the permutation <span class="SimpleMath">p</span> can be computed as <span class="SimpleMath">i</span><code class="code">/</code><span class="SimpleMath">p</span>, without constructing the inverse of <span class="SimpleMath">p</span>.</p>
<p>For arithmetic operations for permutations and their precedence, see <a href="chap31.html#X7A2914307963E370"><span class="RefLink">31.12</span></a>.</p>
<p>In the names of the <strong class="pkg">GAP</strong> functions that deal with permutations, the word "Permutation" is usually abbreviated to "Perm", to save typing. For example, the category test function for permutations is <code class="func">IsPerm</code> (<a href="chap42.html#X7AA69C6686FC49EA"><span class="RefLink">42.1-1</span></a>).</p>
<p><a id="X80F07BE2811D4BAC" name="X80F07BE2811D4BAC"></a></p>
<h4>42.1 <span class="Heading">IsPerm (Filter)</span></h4>
<p>Internally, <strong class="pkg">GAP</strong> stores a permutation as a list of the <span class="SimpleMath">d</span> images of the integers <span class="SimpleMath">1, ..., d</span>, where the "internal degree" <span class="SimpleMath">d</span> is the largest integer moved by the permutation or bigger. When a permutation is read in in cycle notation, <span class="SimpleMath">d</span> is always set to the largest moved integer, but a bigger <span class="SimpleMath">d</span> can result from a multiplication of two permutations, because the product is not shortened if it fixes <span class="SimpleMath">d</span>. The images are stored all as <span class="SimpleMath">16</span>-bit integers or all as <span class="SimpleMath">32</span>-bit integers, depending on whether <span class="SimpleMath">d ≤ 65536</span> or not. For example, if <span class="SimpleMath">m≥ 65536</span>, the permutation <span class="SimpleMath">(1, 2, ..., m)</span> has internal degree <span class="SimpleMath">d=m</span> and takes <span class="SimpleMath">4m</span> bytes of memory for storage. But --- since the internal degree is not reduced --- this means that the identity permutation <code class="code">()</code> calculated as <span class="SimpleMath">(1, 2, ..., m) * (1, 2, ..., m)^{-1}</span> also takes <span class="SimpleMath">4m</span> bytes of storage. It can take even more because the internal list has sometimes room for more than <span class="SimpleMath">d</span> images.</p>
<p>The operation <code class="func">RestrictedPerm</code> (<a href="chap42.html#X7EF8388E7DA8E600"><span class="RefLink">42.5-4</span></a>) reduces the storage degree of its result and therefore can be used to save memory if intermediate calculations in large degree result in a small degree result.</p>
<p>Permutations do not belong to a specific group. That means that one can work with permutations without defining a permutation group that contains them.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,2,3);</span>
(1,2,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,2,3) * (2,3,4);</span>
(1,3)(2,4)
<span class="GAPprompt">gap></span> <span class="GAPinput">17^(2,5,17,9,8);</span>
9
<span class="GAPprompt">gap></span> <span class="GAPinput">OnPoints(17,(2,5,17,9,8));</span>
9
</pre></div>
<p>The operation <code class="func">Permuted</code> (<a href="chap21.html#X7B5A19098406347A"><span class="RefLink">21.20-18</span></a>) can be used to permute the entries of a list according to a permutation.</p>
<p><a id="X7AA69C6686FC49EA" name="X7AA69C6686FC49EA"></a></p>
<h5>42.1-1 IsPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPerm</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Each <em>permutation</em> in <strong class="pkg">GAP</strong> lies in the category <code class="func">IsPerm</code>. Basic operations for permutations are <code class="func">LargestMovedPoint</code> (<a href="chap42.html#X84AA603987C94AC0"><span class="RefLink">42.3-2</span></a>), multiplication of two permutations via <code class="code">*</code>, and exponentiation <code class="code">^</code> with first argument a positive integer <span class="SimpleMath">i</span> and second argument a permutation <span class="SimpleMath">π</span>, the result being the image <span class="SimpleMath">i^π</span> of the point <span class="SimpleMath">i</span> under <span class="SimpleMath">π</span>.</p>
<p><a id="X82069E437D2DF9AA" name="X82069E437D2DF9AA"></a></p>
<h5>42.1-2 IsPermCollection</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPermCollection</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPermCollColl</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>are the categories for collections of permutations and collections of collections of permutations, respectively.</p>
<p><a id="X819628B083B3939B" name="X819628B083B3939B"></a></p>
<h5>42.1-3 PermutationsFamily</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermutationsFamily</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>is the family of all permutations.</p>
<p><a id="X7A21DE5779D21A6D" name="X7A21DE5779D21A6D"></a></p>
<h4>42.2 <span class="Heading">Comparison of Permutations</span></h4>
<p><a id="X7CEC03A9808E2E7C" name="X7CEC03A9808E2E7C"></a></p>
<h5>42.2-1 \=</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \=</code>( <var class="Arg">p1</var>, <var class="Arg">p2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ \<</code>( <var class="Arg">p1</var>, <var class="Arg">p2</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Two permutations are equal if they move the same points and all these points have the same images under both permutations.</p>
<p>The permutation <var class="Arg">p1</var> is smaller than <var class="Arg">p2</var> if <var class="Arg">p1</var> <span class="SimpleMath">≠</span> <var class="Arg">p2</var> and <span class="SimpleMath">i^{<var class="Arg">p1</var>} < i^{<var class="Arg">p2</var>}</span>, where <span class="SimpleMath">i</span> is the smallest point with <span class="SimpleMath">i^{<var class="Arg">p1</var>} ≠ i^{<var class="Arg">p2</var>}</span>. Therefore the identity permutation is the smallest permutation, see also Section <a href="chap31.html#X7B3BC7BA7BB2646D"><span class="RefLink">31.11</span></a>.</p>
<p>Permutations can be compared with certain other <strong class="pkg">GAP</strong> objects, see <a href="chap4.html#X7A274A1F8553B7E6"><span class="RefLink">4.12</span></a> for the details.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,2,3) = (2,3,1);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,2,3) * (2,3,4) = (1,3)(2,4);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,2,3) < (1,3,2); # 1^(1,2,3) = 2 < 3 = 1^(1,3,2)</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">(1,3,2,4) < (1,3,4,2); # 2^(1,3,2,4) = 4 > 1 = 2^(1,3,4,2)</span>
false
</pre></div>
<p><a id="X7BC944F57A04AFF2" name="X7BC944F57A04AFF2"></a></p>
<h5>42.2-2 DistancePerms</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DistancePerms</code>( <var class="Arg">perm1</var>, <var class="Arg">perm2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns the number of points for which <var class="Arg">perm1</var> and <var class="Arg">perm2</var> have different images. This should always produce the same result as <code class="code">NrMovePoints(<var class="Arg">perm1</var>/<var class="Arg">perm2</var>)</code> but some methods may be much faster than this form, since no new permutation needs to be created.</p>
<p><a id="X83A917F67D45C7EA" name="X83A917F67D45C7EA"></a></p>
<h5>42.2-3 SmallestGeneratorPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallestGeneratorPerm</code>( <var class="Arg">perm</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the smallest permutation that generates the same cyclic group as the permutation <var class="Arg">perm</var>. This is very efficient, even when <var class="Arg">perm</var> has large order.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestGeneratorPerm( (1,4,3,2) );</span>
(1,2,3,4)
</pre></div>
<p><a id="X82C255E2821C0721" name="X82C255E2821C0721"></a></p>
<h4>42.3 <span class="Heading">Moved Points of Permutations</span></h4>
<p><a id="X84EF0A697F7A87DC" name="X84EF0A697F7A87DC"></a></p>
<h5>42.3-1 SmallestMovedPoint</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallestMovedPoint</code>( <var class="Arg">perm</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallestMovedPoint</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the smallest positive integer that is moved by <var class="Arg">perm</var> if such an integer exists, and <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) if <var class="Arg">perm</var> is the identity. For <var class="Arg">C</var> a collection or list of permutations, the smallest value of <code class="func">SmallestMovedPoint</code> for the elements of <var class="Arg">C</var> is returned (and <code class="func">infinity</code> (<a href="chap18.html#X8511B8DF83324C27"><span class="RefLink">18.2-1</span></a>) if <var class="Arg">C</var> is empty).</p>
<p><a id="X84AA603987C94AC0" name="X84AA603987C94AC0"></a></p>
<h5>42.3-2 LargestMovedPoint</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestMovedPoint</code>( <var class="Arg">perm</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LargestMovedPoint</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a permutation <var class="Arg">perm</var>, this attribute contains the largest positive integer which is moved by <var class="Arg">perm</var> if such an integer exists, and <code class="code">0</code> if <var class="Arg">perm</var> is the identity. For <var class="Arg">C</var> a collection or list of permutations, the largest value of <code class="func">LargestMovedPoint</code> for the elements of <var class="Arg">C</var> is returned (and <code class="code">0</code> if <var class="Arg">C</var> is empty).</p>
<p><a id="X85E61B9C7A6B0CCA" name="X85E61B9C7A6B0CCA"></a></p>
<h5>42.3-3 MovedPoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MovedPoints</code>( <var class="Arg">perm</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MovedPoints</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the proper set of the positive integers moved by at least one permutation in the collection <var class="Arg">C</var>, respectively by the permutation <var class="Arg">perm</var>.</p>
<p><a id="X85E7B1E28430F49E" name="X85E7B1E28430F49E"></a></p>
<h5>42.3-4 NrMovedPoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrMovedPoints</code>( <var class="Arg">perm</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrMovedPoints</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the number of positive integers that are moved by <var class="Arg">perm</var>, respectively by at least one element in the collection <var class="Arg">C</var>. (The actual moved points are returned by <code class="func">MovedPoints</code> (<a href="chap42.html#X85E61B9C7A6B0CCA"><span class="RefLink">42.3-3</span></a>).)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestMovedPointPerm((4,5,6)(7,2,8));</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPointPerm((4,5,6)(7,2,8));</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPointsPerm((4,5,6)(7,2,8));</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">MovedPoints([(2,3,4),(7,6,3),(5,47)]);</span>
[ 2, 3, 4, 5, 6, 7, 47 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints([(2,3,4),(7,6,3),(5,47)]);</span>
7
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallestMovedPoint([(2,3,4),(7,6,3),(5,47)]);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint([(2,3,4),(7,6,3),(5,47)]);</span>
47
<span class="GAPprompt">gap></span> <span class="GAPinput">LargestMovedPoint([()]);</span>
0
</pre></div>
<p><a id="X79BE80267F4AA2B0" name="X79BE80267F4AA2B0"></a></p>
<h4>42.4 <span class="Heading">Sign and Cycle Structure</span></h4>
<p><a id="X7BE5011B7C0DB704" name="X7BE5011B7C0DB704"></a></p>
<h5>42.4-1 SignPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SignPerm</code>( <var class="Arg">perm</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The <em>sign</em> of a permutation <var class="Arg">perm</var> is defined as <span class="SimpleMath">(-1)^k</span> where <span class="SimpleMath">k</span> is the number of cycles of <var class="Arg">perm</var> of even length.</p>
<p>The sign is a homomorphism from the symmetric group onto the multiplicative group <span class="SimpleMath">{ +1, -1 }</span>, the kernel of which is the alternating group.</p>
<p><a id="X7944D1447804A69A" name="X7944D1447804A69A"></a></p>
<h5>42.4-2 CycleStructurePerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CycleStructurePerm</code>( <var class="Arg">perm</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the cycle structure (i.e. the numbers of cycles of different lengths) of the permutation <var class="Arg">perm</var>. This is encoded in a list <span class="SimpleMath">l</span> in the following form: The <span class="SimpleMath">i</span>-th entry of <span class="SimpleMath">l</span> contains the number of cycles of <var class="Arg">perm</var> of length <span class="SimpleMath">i+1</span>. If <var class="Arg">perm</var> contains no cycles of length <span class="SimpleMath">i+1</span> it is not bound. Cycles of length 1 are ignored.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SignPerm((1,2,3)(4,5));</span>
-1
<span class="GAPprompt">gap></span> <span class="GAPinput">CycleStructurePerm((1,2,3)(4,5,9,7,8));</span>
[ , 1,, 1 ]
</pre></div>
<p><a id="X7B3194EC869D936D" name="X7B3194EC869D936D"></a></p>
<h4>42.5 <span class="Heading">Creating Permutations</span></h4>
<p><a id="X7A9DCFD986958C1E" name="X7A9DCFD986958C1E"></a></p>
<h5>42.5-1 ListPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ListPerm</code>( <var class="Arg">perm</var>[, <var class="Arg">length</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>is a list <span class="SimpleMath">l</span> that contains the images of the positive integers under the permutation <var class="Arg">perm</var>. That means that <span class="SimpleMath">l</span><code class="code">[</code><span class="SimpleMath">i</span><code class="code">]</code> <span class="SimpleMath">= i</span><code class="code">^</code><var class="Arg">perm</var>, where <span class="SimpleMath">i</span> lies between 1 and the largest point moved by <var class="Arg">perm</var> (see <code class="func">LargestMovedPoint</code> (<a href="chap42.html#X84AA603987C94AC0"><span class="RefLink">42.3-2</span></a>)).</p>
<p>An optional second argument specifies the length of the desired list.</p>
<p><a id="X78D611D17EA6E3BC" name="X78D611D17EA6E3BC"></a></p>
<h5>42.5-2 PermList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PermList</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the permutation <span class="SimpleMath">π</span> that moves points as described by the list <var class="Arg">list</var>. That means that <span class="SimpleMath">i^π =</span> <var class="Arg">list</var><code class="code">[</code><span class="SimpleMath">i</span><code class="code">]</code> if <span class="SimpleMath">i</span> lies between <span class="SimpleMath">1</span> and the length of <var class="Arg">list</var>, and <span class="SimpleMath">i^π = i</span> if <span class="SimpleMath">i</span> is larger than the length of the list <var class="Arg">list</var>. <code class="func">PermList</code> will return <code class="keyw">fail</code> if <var class="Arg">list</var> does not define a permutation, i.e., if <var class="Arg">list</var> is not dense, or if <var class="Arg">list</var> contains a positive integer twice, or if <var class="Arg">list</var> contains an integer not in the range <code class="code">[ 1 .. Length( <var class="Arg">list</var> ) ]</code>. If <var class="Arg">list</var> contains non-integer entries an error is raised.</p>
<p><a id="X8087DCC780B9656A" name="X8087DCC780B9656A"></a></p>
<h5>42.5-3 MappingPermListList</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MappingPermListList</code>( <var class="Arg">src</var>, <var class="Arg">dst</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">src</var> and <var class="Arg">dst</var> be lists of positive integers of the same length, such that neither may contain an element twice. <code class="func">MappingPermListList</code> returns a permutation <span class="SimpleMath">π</span> such that <var class="Arg">src</var><code class="code">[</code><span class="SimpleMath">i</span><code class="code">]^</code><span class="SimpleMath">π =</span> <var class="Arg">dst</var><code class="code">[</code><span class="SimpleMath">i</span><code class="code">]</code>. The permutation <span class="SimpleMath">π</span> fixes all points larger than the maximum of the entries in <var class="Arg">src</var> and <var class="Arg">dst</var>. If there are several such permutations, it is not specified which of them <code class="func">MappingPermListList</code> returns.</p>
<p><a id="X7EF8388E7DA8E600" name="X7EF8388E7DA8E600"></a></p>
<h5>42.5-4 RestrictedPerm</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedPerm</code>( <var class="Arg">perm</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RestrictedPermNC</code>( <var class="Arg">perm</var>, <var class="Arg">list</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">RestrictedPerm</code> returns the new permutation that acts on the points in the list <var class="Arg">list</var> in the same way as the permutation <var class="Arg">perm</var>, and that fixes those points that are not in <var class="Arg">list</var>. The resulting permutation is stored internally of degree given by the maximal entry of <var class="Arg">list</var>. <var class="Arg">list</var> must be a list of positive integers such that for each <span class="SimpleMath">i</span> in <var class="Arg">list</var> the image <span class="SimpleMath">i</span><code class="code">^</code><var class="Arg">perm</var> is also in <var class="Arg">list</var>, i.e., <var class="Arg">list</var> must be the union of cycles of <var class="Arg">perm</var>.</p>
<p><code class="func">RestrictedPermNC</code> does not check whether <var class="Arg">list</var> is a union of cycles.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ListPerm((3,4,5));</span>
[ 1, 2, 4, 5, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PermList([1,2,4,5,3]);</span>
(3,4,5)
<span class="GAPprompt">gap></span> <span class="GAPinput">MappingPermListList([2,5,1,6],[7,12,8,2]);</span>
(1,8,5,12,11,10,9,6,2,7,4,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">RestrictedPerm((1,2)(3,4),[3..5]);</span>
(3,4)
</pre></div>
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