<h3>7 <span class="Heading">Higher level methods for pcp-groups</span></h3>
<p>This is a description of some higher level functions of the <strong class="pkg">Polycyclic</strong> package of GAP 4. Throughout this chapter we let <var class="Arg">G</var> be a pc-presented group and we consider algorithms for subgroups <var class="Arg">U315
<h4>7.1 <span class="Heading">Subgroup series in pcp-groups</span></h4>
<p>Many algorithm for pcp-groups work by induction using some series through the group. In this section we provide a number of useful series for pcp-groups. An <em]><Example>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EfaSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a normal series of <var class="Arg">U</var> with elementary or free abelian factors.</p>
<ivclass=func " width=100%>tr><d ="tdleft"code class""&8227 SemiSimpleEfaSeries</ode>(<var class"">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns an efa series of <var class="Arg">U</var> such that every factor in the series is semisimple as a module for <var class="Arg">U</var> over a finite field or over the rationals.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RefinedDerivedSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>the derived series of <var class="Arg">U</var> refined to an efa series such that in each abelian factor of the derived series the free abelian factor is at the top.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><P/
<p>the derived series of <var class="Arg">U</var> refined to an efa series such that in each abelian factor of the derived series the free abelian factor is at the bottom.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LowerCentralSeriesOfGroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>the lower central series of <var class="Arg">U</var>. If <var class="Arg">U</var> does not have a largest nilpotent quotient group, then this function may not terminate.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; UpperCentralSeriesOfGroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>the upper central series of <var class="Arg">U</var>. This function always terminates, but it may terminate at a proper subgroup of <var class="Arg">U</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TorsionByPolyEFSeries</code>( <var class="Arg">U</var> )</td><tdclass="tdright">( function )</td></tr></table></div>
<p>returns an efa series of <var class="Arg">U</var> such that all torsion-free factors are at the top and all finite factors are at the bottom. Such a series might not exist for <var class="Arg">U</var> and in this case the function returns> x ->RatioOfNongenerationTransPermGroup(g,2A x );
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(5);</span>ap> minMinimum( )java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
Pcp-group with orders [ 2, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Igs(G);</span>
[ g1, g2, g3, g4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpSeries(G);</span>
[ Pcp-group with orders [ 2, 0, 0, 0 ],
Pcp-group with orders [ 0, 0, 0 ],
Pcp-group with orders [ 0, 0 ],
Pcp-group with orders [ 0 ],
Pcp-group with orders [ ] ]
<p>Algorithms for pcp-groups often use an efa series of <span class="SimpleMath">\(G\)</span> and work down over the factors of this series. Usually, pcp's of the factors are more useful than the actual factors. Hence we provide the following.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PcpsBySeries</code>( <var class="Arg">ser</var>[, <var class="Arg">flag</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a list of pcp's corresponding to the factors of the series. If the parameter <var class="Arg">flag</var> is present and equals the string <q>snf</q>, then each pcp corresponds to a decomposition of the abelian groups into direct factors.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PcpsOfEfaSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of pcps corresponding to an efa series of <var class="Arg">U</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 29 out of bounds for length 0
Pcp-group with orders [ 2, 0, 0, 0 ]
<h4>7.2 <span class="Heading">Orbit stabilizer methods for pcp-groups</span></h4>
<p>Let <var class="Arg">U</var> be a pcp-group which acts on a set <span class="SimpleMath">\(\Omega\)</span>. One of the fundamental problems in algorithmic group theory is the determination of orbits and stabilizers of points in <span class="SimpleMath">\(\Omega\)</span> under the action of <var class="Arg">U</var>. We distinguish two cases: the case that all considered orbits are finite and the case that there are infinite orbits. In the latter case, an orbit cannot be listed and a description of the orbit and its corresponding stabilizer is much harder to obtain.</p>
<p>If the considered orbits are finite, then the following two functions can be applied to compute the are larger than<>13<M.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PcpOrbitStabilizer</code>( <var class="Arg">point</var>, <var class="Arg">gens</var>, <var class="Arg">acts</var>, <var class="Arg">oper</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PcpOrbitsStabilizers</code>( <var class="Arg">points</var<P/>
<p>The input <var class="Arg">gens</var> can be an igs or a pcp of a pcp-group <var class="Arg">U</var>. The elements in the list <var class="Arg">gens</var> act as the elements in the list <var class="Arg">acts</var> via the function <var class="Arg">oper</var> on the given points; that is, <var class="Arg">oper( point, acts[i] )</var> applies the <span class="SimpleMath">\(i\)</span>th generator to a given point. Thus the group defined by <var class="Arg">acts</var> must be a homomorphic image of the group defined by <var class="Arg">gens</var>. The first function returns a record containing the orbit as component 'orbit' and and igs for the stabilizer as component '
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><codegap>Minimum(bad);
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ConjugacyIntegralAction</code>( <var class="Arg">U</var>, <varclass="Arg">mats</var>, <var class="Arg">B</var>, <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>The first function computes the normalizer in <var class="Arg">U</var> of the lattice with the basis <var class="Arg">B</var>, where the pcp group <var class="Arg">U</var> acts via <var class="Arg">mats</var> on an integral space and <var class="Arg">B</var> is a subspace of this integral space. The second functions checks whether the two lattices with the bases <var class="Arg">B</var> and <var class="Arg
<div class="example"><pre>
# get a pcp group and a free abelian normal subgroup
<span class="GAPprompt">gap></span> <span class="GAPinput">G := ExamplesOfSomePcpGroups(8);</span>
Pcp-group with orders [ 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">efa := EfaSeries(G);</span>
[ Pcp-group with orders [ 0, 0, 0, 0, 0 ],
Pcp-group with orders [ 0, 0, 0, 0 ],
Pcp-group with orders [ 0, 0, 0 ],
Pcp-group with orders [ ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">N := efa[3];</span>
Pcp-group with orders [ 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsFreeAbelian(N);</span>
true
# take an arbitrary vector and compute its stabilizer
<span class="GAPprompt">gap></span> <span class="GAPinput">StabilizerIntegralAction(G,mats, [2,3,4]);</span>
Pcp-group with orders [ 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Igs(last);</span>
[ g1, g3, g4, g5 ]
# check orbits with some other vectors
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 0
rec( stab := Pcp-group with orders [ 0, 0, 0, 0 ], prei := g2 )
# compute the orbit of a subgroup of Z^3 under the action of G
<span class="GAPprompt">gap></span> <span class="GAPinput">NormalizerIntegralAction(G, mats, [[1,0,0],[0,1,0]]);</span>
Pcp-group with orders [ 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Igs(last);</span>
[ g1, g2^2, g3, g4, g5 ]
</pre></div>
<h4>7.3 <span class="Heading">Centralizers, Normalizers and Intersections</span></h4>
<p>In this section we list a number of operations for which there are methods installed to compute the corresponding features in polycyclic groups.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Centralizer</code>( <var class="Arg">U</var>, <var class="Arg">g</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">&#an style='color: green'>8227; IsConjugate</code>( <var class="Arg">U</var>, <var class="Arg">g</var>, <var class="Arg">h</We in
<p>These functions solve the conjugacy problem for elements in pcp-groups and they can be used to compute centralizers. The first method returns a subgroup of the given group <var class="Arg">U</var>, the second method either returns a conjugating element or false if no such element exists.</p>
<p>The methods are based on the orbit stabilizer algorithms described in <a href="chapBib_mj.html#biBEOs01">[EO02]</a>. For nilpotent groups, an algorithm to solve the conjugacy problem for elements is described in <a href="chapBib_mj.html#biBSims94">[Sim94]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Centralizer</code>( <var class="Arg">U</var>, <var class="Arg">V</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsConjugate</code>( <var class="Arg">U</var>, <var class="Arg">V</var>, <var class="Arg">W</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>These three functions solve the conjugacy problem for subgroups and compute centralizers and normalizers of subgroups. The first two functions return subgroups of the input group <var class="Arg">U</var>, the third function returns a conjugating element or false if no such element exists.</p>
<p>The methods are based on the orbit stabilizer algorithms described in <a href="chapBib_mj.html#biBEic02">[Eic02]</a>. For nilpotent groups, an algorithm to solve the conjugacy problems for subgroups is described in <a href="chapBib_mj.html#biBLo98">[Lo98b]</a>.</p>
<p><a id="X851069107CACF98E" name="java.lang.StringIndexOutOfBoundsException: Index 50 out of bounds for length 27
<h5>7.3-3 Intersection</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&gap>; (g, ,2 ,,20
<p>A general method to compute intersections of subgroups of a pcp-group is described in <a href="chapBib_mj.html#biBEic01b">[Eic01a]</a>, but it is not yet implemented here. However, intersections of subgroups <span class="SimpleMath">\(U, N \leq G\)</span> can be computed if <span class="SimpleMath">\(N\)</span> is normalising <span class="SimpleMath">\(U\)</span>. See <a href="chapBib_mj.html#biBSims94">[Sim94]</a> for an outline of the algorithm.</p>
<p>There are various finite subgroups of interest in polycyclic groups. See <a href="chapBib_mj.html#biBEic00">[Eic00]</a> for a description of the algorithms underlying the functions in this section.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TorsionSubgroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>If the set of elements of finite order forms a subgroup, then we call it the <em>torsion subgroup</em>. This function determines the torsion subgroup of <var class="Arg">U</var>, if it exists, and returns fail otherwise. Note that a torsion subgroup does always exist if <var class="Arg">U</var> is nilpotent.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NormalTorsionSubgroup</code>( <var class="Arg">U</var> )</td><tdclass="tdright">( attribute )</td></tr></table></div>
<p>Each polycyclic groups has a unique largest finite normal subgroup. This function computes it for <var class="Arg">U</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiniteSubgroupClasses</code>( <var class="Arg">U</var> )</td><tdclass="tdright">( attribute )</td></tr></table></div>
<p>There exist only finitely many conjugacy classes of finite subgroups in a polycyclic group <var class="Arg">U</var> and this function can be used to compute them. The algorithm underlying this function proceeds by working down a normal series of <var class="Arg">U</var> with elementary or free abelian factors. The following function can be used to give the algorithm a specific series.</p>
<h4>7.5 <span class="Heading">Subgroups of finite index and maximal subgroups</span></h4>
<p>Here we outline functions to determine various types of subgroups of finite index in polycyclic groups. Again, see <a href="chapBib_mj.html#biBEic00">[Eic00]</a> for a description of the algorithms underlying the functions in this section. Also, we refer to <a href="chapBib_mj.html#biBLo99">[Lo98a]</a> for an alternative approach.</p>
<h5>7.5-java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MaximalSubgroupClassesByIndex</code>( <var class="Arg">U</var>, <var class="Arg">p</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Each maximal subgroup of a polycyclic group <var class="Arg">U</var> has <var class="Arg">p</var>-power index for some prime <var class="Arg">p</var>. This function can be used to determine the conjugacy classes of all maximal subgroups of <var class="Arg">p</var>-power index for a given prime <var class="Arg">p</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LowIndexSubgroupClasses</code>( <var class="Arg">U</var>, <varclass="Arg">n</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>There are only finitely many subgroups of a given index in a polycyclic group <var class="Arg">U</var>. This function computes conjugacy classes of all subgroups of index <var class="Arg">n</var> in <var class="Arg">U</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LowIndexNormalSubgroups</code>( <var class="Arg">U</var>, <varclass="Arg">n</var>
<p>This function computes the normal subgroups of index <var class="Arg">n</var> in <var class="Arg">U</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NilpotentByAbelianNormalSubgroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a normal subgroup <var class="Arg">N</var> of finite index in <var class="Arg">U</var> such that <var class="Arg">N</var> is nilpotent-by-abelian. Such a subgroup exists in every polycyclic group and this function computes such a subgroup using LowIndexNormal. However, we note that this function is not very efficient and the function NilpotentByAbelianByFiniteSeries may well be more efficient on this task.</p>
<h4>7.6 <span class="Heading">Further attributes for pcp-groups based on the Fitting subgroup</span></h4>
<p>In this section we provide a variety of other attributes for pcp-groups. Most of the methods below are based or related to the Fitting subgroup of the given group. We refer to <a href="chapBib_mj.html#biBEic01">[Eic01b]</a> for a description of the underlying methods.</p>
<div class="func"><table class="func" width
<p>returns the Fitting subgroup of <var class="Arg">U</var>; that is, the largest nilpotent normal subgroup of <var class="Arg">U</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FCCentre</code>( <var class="Arg">U</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>returns the FC-centre of <var class="Arg">U</var>; that is, the subgroup containing all elements having a finite conjugacy class in <var class="Arg">U</var>.</p>
<p
<h5>7.6-5 PolyZNormalSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PolyZNormalSubgroup</code>( <var class="Arg">U</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a normal subgroup <var class="Arg">N</var> of finite index in <var class="Arg">U</var>, such that <var class="Arg">N</var> has a polycyclic series with infinite factors onlygjava.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 36
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NilpotentByAbelianByFiniteSeries</code>( <var class="Arg">U</var> )</td><td class="tdright">( functiongap (java.lang.StringIndexOutOfBoundsException: Range [58, 56) out of bounds for length 76
<p>returns a normal series <span class="SimpleMath">\(1 \leq F \leq A \leq U\)</span> such that <spanclass="SimpleMath">\(F\)</span> is nilpotent, <span class="SimpleMath">\(A/F\)</span> is abelian and <span class="SimpleMath">\(U/A\)</span> is finite. This series java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 44
<p>Below we introduce a function which computes orbit and stabilizer using a random method. This function tries to approximate the orbit and the stabilizer, but the returned orbit or stabilizer may be incomplete. This function is used in the random methods to compute normalizers and centralizers. Note that deterministic
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; SchurExtension</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a polycyclic group with a polycyclic generating sequence consisting of <span class="SimpleMath">\(n\)</span> elements. This function computes the largest central extension <var class="Arg">H</var> of <var class="Arg">G</var> such that <var class="Arg">H</var> is generated by <span class="SimpleMath">\(n\)</span> elements. If <span class="SimpleMath">\(F/R\)</span> is the underlying polycyclic presentation for <var class="Arg">G</var>, then <var class="Arg">H</var> is isomorphic to <span class="SimpleMath">\(F/[R,F]\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DihedralPcpGroup( 0 );</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Centre( G );</span>
Pcp-group with orders [ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">H := SchurExtension( G );</span>
Pcp-group with orders [ 2, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Centre( H );</span>
Pcp-group with orders [ 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">H/Centre(H);</span>
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Subgroup( H, [H.1,H.2] ) = H;</span>
true
</pre></div>
<p>There is a crossed pairing from <var class="Arg">G</var> into <span class="SimpleMath">\((G^{*})'\)</span> which can be defined via this epimorphism:</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; SchurCover</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes a Schur covering group of the polycyclic group <var class="Arg">gap> PositionsProperty( approx, x -> x > 1/ )
<p>If <var class="Arg">G</var> is given by a presentation <span class="SimpleMath">\(F/R\)</span>, then <var class="Arg">M</var> is isomorphic to the subgroup <span class="SimpleMath">\(R \cap [F,F] / [R,F]\)</span>. Let <span class="SimpleMath">\(C\)</span[2]
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; AbelianInvariantsMultiplier</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of the abelian invariants of the Schur multiplier of G.</p>
<p>Note that the Schur multiplicator of a polycyclic group is a finitely generated abelian group.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; NonAbelianExteriorSquareEpimorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the epimorphism of the non-abelian exterior square of a polycyclic group <var class="Arg">G</var> onto the derived group of <var class="Arg">G</var>. The non-abelian exterior square can be defined as the derived subgroup of a Schur cover of <var class="Arg">G</var>. The isomorphism type of the non-abelian exterior square is unique despite the fact that the isomorphism type of a Schur cover of a polycyclic groups need not be unique. The derived group of a Schur cover has a natural projection onto the derived group of <var class="Arg">G</var> which is what the function returns.</Instatement&bsp() onlythe that
<p>The kernel of the epimorphism is isomorphic to the Schur multiplicator of <var class="Arg">G</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; NonAbelianExteriorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes the non-abelian exterior square of a polycyclic group <var class="Arg">G</var>. See the explanation for <code class="code">NonAbelianExteriorSquareEpimorphism</code>. The natural projection of the non-abelian exterior square onto the derived group of <var class="Arg">G</var> is stored in the component <code class="code">!.epimorphism</code>.</p>
<p>There is a crossed pairing from <span class="SimpleMath">\(G\times G\)</span> into <span class="SimpleMath">\(G\wedge G\)</span>. See the function <code class="code">SchurExtensionEpimorphism</code> for details. The crossed pairing is stored in gap d: 10;
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; NonAbelianTensorSquareEpimorphism</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns for a polycyclic group <var class="Arg">G</var> the projection of the non-abelian tensor square <span class="SimpleMath">\(G\otimes G\)</span> onto the non-abelian exterior square <span class="SimpleMath">\(G\wedge G\)</span>. The range of that epimorphism has the component <code class="code">!.epimorphism</code> set to the projection of the non-abelian exterior square onto the derived group of <var class="Arg">G</var>. See also the function <code class="code">NonAbelianExteriorSquare</code>.</p>
<p>With the result of this function one can compute the groups in the commutative diagram at the beginning of the paper <a href="chapBib_mj.html#biBEickNickel07">[EN08]</a>. The kernel of the returned epimorphism is the group <span class="SimpleMath">\(\nabla(G)\)</span>. The kernel of the composition of this epimorphism and the above mention projection onto <span class="SimpleMath">\(G'\)</span> is the group <span class="SimpleMath">\(J(G)\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; NonAbelianTensorSquare</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes for a polycyclic group <var class="Arg">G</var> the non-abelian tensor square <span class="SimpleMath">\(G\otimes G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := AlternatingGroup( IsPcGroup, 4);</span>
<pc group of size 12 with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">PcGroupToPcpGroup( G );</span>
Pcp-group with orders [ 3, 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">NonAbelianTensorSquare( last );</span>
Pcp-group with orders [ 2, 2, 2, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PcpGroupToPcGroup( last );</span>
<pc group of size 24 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">DirectFactorsOfGroup( last );</span>
[ Group([ f1, f2, f3 ]), Group([ java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">List( last, Size );</span>
[ 8, 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( last2[1] );</span>
[ 8, 4 ] # the quaternion group of Order 8
<span class="GAPprompt">gap></span<--%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -java.lang.StringIndexOutOfBoundsException: Index 79 out of bounds for length 79
Pcp-group with orders [ 2, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ten := NonAbelianTensorSquare( G );</span>
Pcp-group with orders [ 0, 2, 2, 2 ]
spanclass="APprompt">gap&t;/><spanclass"">sAbelian ten )<span>
true
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; NonAbelianExteriorSquarePlusEmbedding</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns an embedding from the non-abelian exterior square <span class="SimpleMath">\(G\wedge G\)</span> into an extensions of <span class="SimpleMath">\(G\wedge G\)</span> by <span class="SimpleMath">\(G\times G\)</span>. For the significance of the group see the paper <a href="chapBib_mj.html#biBEickNickel07">[EN08]</a>. The range of the epimorphism is the group <span class="SimpleMath">\(\tau(G)\)</span> in that paper.</p>
<div class="func"><table class=
<p>returns the group <span class="SimpleMath">\(\nu(G)\)</span> in <a href="java.lang.StringIndexOutOfBoundsException: Index 83 out of bounds for length 43
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&an style='color:green'>#8227; SchurCovers</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <var class="Arg">G</var> be a finite <span class="SimpleMath">\(p\)</span>-group defined as a pcp group. This function returns a complete and irredundant set of isomorphism types of Schur covers of <var class="Arg">G</var>. The algorithm implements a method of Nickel's Phd Thesis.</p>
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