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<a id="X83F7986D84F33F6C" name="X83F7986D84F33F6C"></a>
<div class="ChapSects"><a href="chap5.html#X83F7986D84F33F6C">5 Local Functions in the PERMUT Package</a>
<div class="ContSect"> <a href="chap5.html#X7E8389FD7DE9769C">5.1 A Local Function for Supersolubility</a>

<div class="ContSSBlock">
 <a href="chap5.html#X7B9F19A37F4A5B35">5.1-1 IsPSupersolvable</a>
</div></div>
<div class="ContSect"> <a href="chap5.html#X7C4B2A9A7F2D7E0B">5.2 Local functions for T-groups, PT-groups, and PST-groups</a>

<div class="ContSSBlock">
 <a href="chap5.html#X780657F179FFB7EE">5.2-1 IsCp</a>
 <a href="chap5.html#X82953B4878F88327">5.2-2 IsXp</a>
 <a href="chap5.html#X8304885C78E4A584">5.2-3 IsYp</a>
</div></div>
<div class="ContSect"> <a href="chap5.html#X8540D4B679519F8B">5.3 Auxiliary Functions for T-groups, PT-groups, and PST-groups</a>

<div class="ContSSBlock">
 <a href="chap5.html#X8775CAFD80D3FC02">5.3-1 IsAbCp</a>
 <a href="chap5.html#X7A6372D3871E8E55">5.3-2 IsDedekindSylow</a>
 <a href="chap5.html#X812C3A017F534CC0">5.3-3 IwasawaTripleWithSubgroup</a>
 <a href="chap5.html#X7E86B21086879C48">5.3-4 IwasawaTriple</a>
 <a href="chap5.html#X8121A92F7EABEA0C">5.3-5 IsIwasawaSylow</a>
</div></div>
</div>

<h3>5 Local Functions in the PERMUT Package</h3>

In the study of permutability, the usage of local characterisations has become a useful tool to describe the classes of T-groups, PT-groups, and PST-groups. In this chapter we present some local characterisations of these classes and some functions which allow to check whether or not a group given in GAP satisfies these conditions.

A local description of group-theoretical property consists of expressing it as the conjunction of some properties depending on a prime p, usually related to the behaviour of p-elements, p-subgroups, or p-chief factors, for all primes p.

<a id="X7E8389FD7DE9769C" name="X7E8389FD7DE9769C"></a>

<h4>5.1 A Local Function for Supersolubility</h4>

The GAP library does not contain methods to check whether a group G is p-supersoluble, where p is a prime number. We include such a function in the PERMUT package.

<a id="X7B9F19A37F4A5B35" name="X7B9F19A37F4A5B35"></a>

<h5>5.1-1 IsPSupersolvable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPSupersolvable</code>( <var class="Arg">G</var>, <var class="Arg">p</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">‣ IsPSupersoluble</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
This function returns <code class="keyw">true</code> if the group <var class="Arg">G</var> is p-supersoluble, and <code class="keyw">false</code> otherwise, where <var class="Arg">p</var> is a prime number. This function is not defined in GAP. The method we have implemented for finite groups includes checking whether the group is supersoluble (in this case, it must return <code class="keyw">true</code>). If the group is not soluble, it computes a chief series and checks whether all chief factors have order p or have order not divisible by p.

<div class="example"><pre>
gap> g:=Group((1,2,3,4,5,6,7), (8,9,10,11,12,13,14), (15,16,17,18,19,20,21),
> (22,23,24,25,26,27,28), (29,30,31,32,33,34,35),
> (1,8,15,22,29)(2,9,16,23,30)(3,10,17,24,31)(4,11,18,25,32)(5,12,19,26,
> 33)(6,13,20,27,34)(7,14,21,28,35),
> (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)); #C7 wr S5
<permutation group with 7 generators>
gap> IsPSupersolvable(g,7);
false
gap> IsPSupersolvable(g,11);
true
</pre></div>

<div class="example"><pre>
gap> g:=DirectProduct(PSL(2,7),
> Group((1,2,3,4,5,6,7,8,9,10,11), (2,5,6,10,4)(3,9,11,8,7)));
Group([ (3,7,5)(4,8,6), (1,2,6)(3,4,8), (9,10,11,12,13,14,15,16,17,18,19),
 (10,13,14,18,12)(11,17,19,16,15) ])
gap> IsPNilpotent(g,5);
true
gap> IsPNilpotent(g,11);
false
gap> IsPSupersolvable(g,11);
true
gap> IsPNilpotent(g,3);
false
</pre></div>

<a id="X7C4B2A9A7F2D7E0B" name="X7C4B2A9A7F2D7E0B"></a>

<h4>5.2 Local functions for T-groups, PT-groups, and PST-groups</h4>

The following functions correspond to local description of the classes of soluble T-groups, PT-groups, and PST-groups. Most of the known useful local characterisations of these classes of groups can be seen to be equivalent to one of them, either in the universe or all finite groups or in the universe of all finite p-soluble groups. By a local characterisation of a group-theoretical property R we mean a group-theoretical property R_p for each prime p such that a group satisfies R if and only if it satisfies R_p for all primes p.

<a id="X780657F179FFB7EE" name="X780657F179FFB7EE"></a>

<h5>5.2-1 IsCp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCp</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
This function returns <code class="keyw">true</code> if the group <var class="Arg">G</var> satisfies the property C_p, where p is a prime number, and <code class="keyw">false</code> otherwise.

A group G satisfies C_p when every subgroup H of a Sylow p-subgroup P of G is normal in the corresponding Sylow normaliser N_G(P). This property was introduced by Robinson in <a href="chapBib.html#biBRobinson68">[Rob68]</a>. A group G is a soluble PST-group if and only if it satisfies C_p for all primes p.

<div class="example"><pre>
gap> g:=AlternatingGroup(5);
Alt( [ 1 .. 5 ] )
gap> IsCp(g,3);
true
gap> IsCp(g,5);
true
gap> IsCp(g,7);
true
gap> IsCp(g,2);
false
gap> g:=SmallGroup(200,44); # semidirect product of Q8 with C5xC5
<pc group of size 200 with 5 generators>
gap> IsCp(g,5);
false
gap> IsCp(g,2);
true
</pre></div>

<a id="X82953B4878F88327" name="X82953B4878F88327"></a>

<h5>5.2-2 IsXp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsXp</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
This function returns <code class="keyw">true</code> if <var class="Arg">G</var> satisfies X_p, where p is a prime, and <code class="keyw">false</code> otherwise.

A group G satisfies X_p when for every subgroup H of a Sylow p-subgroup P of G, H is permutable in N_G(P). This property was introduced by Beidleman, Brewster, and Robinson in <a href="chapBib.html#biBBeidlemanBrewsterRobinson99">[BBR99]</a>. A group G is a soluble PT-group if and only if G satisfies X_p for all primes p.

<div class="example"><pre>
gap> g:=SmallGroup(189,7);
<pc group of size 189 with 4 generators>
gap> IsXp(g,3);
true
gap> IsXp(g,7);
true
gap> IsPTGroup(g);
true
gap> IsCp(g,3);
false
gap> IsTGroup(g);
false
</pre></div>

<a id="X8304885C78E4A584" name="X8304885C78E4A584"></a>

<h5>5.2-3 IsYp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsYp</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
This function returns <code class="keyw">true</code> if <var class="Arg">G</var> satisfies Y_p, where p is a prime, and <code class="keyw">false</code> otherwise.

A group G satisfies Y_p when for every two subgroups H and K with H≤ K, H is S-permutable in N_G(P). This property was introduced by Ballester-Bolinches and Esteban-Romero in <a href="chapBib.html#biBBallesterEsteban02-sylper1">[BBER02]</a>. A group G is a soluble PST-group if and only if G satisfies Y_p for all primes p.

<div class="example"><pre>
gap> g:=SmallGroup(200,43); # semidirect product of D8 with C5xC5
<pc group of size 200 with 5 generators>
gap> IsCp(g,2);
false
gap> IsXp(g,2);
false
gap> IsYp(g,2);
true
gap> g:=Group((1,2,3)(4,5,6),(1,2));
Group([ (1,2,3)(4,5,6), (1,2) ])
gap> IsYp(g,3);
false
gap> IsYp(g,2);
true
</pre></div>

<a id="X8540D4B679519F8B" name="X8540D4B679519F8B"></a>

<h4>5.3 Auxiliary Functions for T-groups, PT-groups, and PST-groups</h4>

The following functions are used to check whether or not a group is a soluble T-group, PT-group, or PST-group.

<a id="X8775CAFD80D3FC02" name="X8775CAFD80D3FC02"></a>

<h5>5.3-1 IsAbCp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAbCp</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
This function returns <code class="keyw">true</code> if <var class="Arg">G</var> has an abelian Sylow <var class="Arg">p</var>-subgroup <var class="Arg">p</var> such that every subgroup of P is normal in the Sylow normaliser N_G(P), and <code class="keyw">false</code> otherwise.

This function is used to characterise soluble PST-groups: a group G is a soluble PST-group if and only if G satisfies Y_p for all primes p, and a group G satisfies Y_p if and only if G is p-nilpotent or G has an abelian Sylow p-subgroup and satisfies C_p. A group G satisfies C_p if and only if every subgroup of a Sylow p-subgroup P of G is normal in the Sylow normaliser N_G(P). Therefore this function checks whether G has an abelian Sylow p-subgroup and G satisfies C_p.

<div class="example"><pre>
gap> g:=AlternatingGroup(5);
Alt( [ 1 .. 5 ] )
gap> IsAbCp(g,5);
true
</pre></div>

<a id="X7A6372D3871E8E55" name="X7A6372D3871E8E55"></a>

<h5>5.3-2 IsDedekindSylow</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsDedekindSylow</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
This function returns <code class="keyw">true</code> if a Sylow <var class="Arg">p</var>-subgroup of <var class="Arg">G</var> is Dedekind, else it returns <code class="keyw">false</code>.

A group G is Dedekind when every subgroup of G is normal. If p is a prime, a Dedekind p-group (see for example 2.3.12 in <a href="chapBib.html#biBSchmidt94">[Sch94]</a>) is abelian or a direct product of a quaternion group of order 8 and an elementary abelian 2-group. Obviously, a p-group is Dedekind if and only if it is a T-group.

The algorithm used in this function to test whether a non-abelian 2-group satisfies this condition checks that the Frattini subgroup of a Sylow 2-subgroup P of G has order 2 and that the centre of P has exponent 2 and index 4. In this case, it computes the natural epimorphism from P to P/Z(P) and it checks that the preimages of the generators of P/Z(P) under the natural epimorphism have order 4. If all these conditions hold, then the Sylow 2-subgroup is Dedekind, otherwise it is not.

This function tries to set the property <code class="func">IsTGroup</code> (<a href="chap4.html#X7930E5AD817BE4B3">4.2-1</a>) to <code class="keyw">true</code> or <code class="keyw">false</code> for the Sylow p-subgroup.

<div class="example"><pre>
gap> g:=DirectProduct(SmallGroup(8,4),CyclicGroup(5));
<pc group of size 40 with 4 generators>
gap> IsDedekindSylow(g,2);
true
</pre></div>

<a id="X812C3A017F534CC0" name="X812C3A017F534CC0"></a>

<h5>5.3-3 IwasawaTripleWithSubgroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IwasawaTripleWithSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">X</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
This function returns an Iwasawa triple for a <var class="Arg">p</var>-group <var class="Arg">G</var> such that <var class="Arg">X</var> is a member of it, if such a triple exists, and <code class="keyw">fail</code> otherwise. This function is used as an auxiliary function to compute an Iwasawa triple for a group <var class="Arg">G</var>.

An Iwasawa triple for a p-group G is a triple (X,b,s) such that X is an abelian normal subgroup of G with cyclic quotient, b is a generator of a supplement to X in G, and b induces a power automorphism in X of the form x-> x^1+p^s. A theorem of Iwasawa states that a p-group G has a modular subgroup lattice (or, equivalently, G has all subgroups permutable) if and only if G is a direct product of a quaternion group of order 8 and an elementary abelian 2-group or G has an Iwasawa triple (X,b,s) with s ≥ 2.

The construction of the Iwasawa triple takes a generator b of a cyclic supplement to X in G. Then we consider a generator a of X of the largest possible order and find an element c of ⟨ b ⟩ and an element s such that a^c = a^1+p^s. If such an element does not exist, the function returns <code class="keyw">fail</code>. For this element, it checks whether for all generators t of X, the equality t^c = t^1+p^s holds. If this holds, it returns the triple (X, c, s); otherwise it returns <code class="keyw">fail</code>.

<div class="example"><pre>
gap> e:=ExtraspecialGroup(27,9);
<pc group of size 27 with 3 generators>
gap> IwasawaTripleWithSubgroup(e,Subgroup(e,[e.1,e.3]),3);
[ Group([ f1, f3 ]), f2, 1 ]
</pre></div>

<a id="X7E86B21086879C48" name="X7E86B21086879C48"></a>

<h5>5.3-4 IwasawaTriple</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IwasawaTriple</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
This function computes an Iwasawa triple for the p-group <var class="Arg">G</var>, if it exists. If <var class="Arg">G</var> is not Iwasawa, the function returns <code class="keyw">fail</code>. If <var class="Arg">G</var> is a direct product of an elementary abelian 2-group and a quaternion group of order 8, it returns an empty list. If <var class="Arg">G</var> is Iwasawa, then the function returns an Iwasawa triple for <var class="Arg">G</var>. An Iwasawa triple for a group <var class="Arg">G</var> is a triple (X, b, s) where X is an abelian normal subgroup of G such that G/X is cyclic, b is a generator of a cyclic supplement to X in G, and s is an integer such that for all x ∈ X, x^b = x^1+p^s. A theorem of Iwasawa states that a p-group G has a modular subgroup lattice (or, equivalently, G has all subgroups permutable) if and only if G is a direct product of an elementary abelian 2-group and a quaternion group of order 8 or G has an Iwasawa triple (X,b,s) with s ≥ 2 if p = 2.

The method followed to find an Iwasawa triple for non-abelian non-Dedekind groups begins with the whole group G. If the group is abelian, it returns the Iwasawa triple (G,1,log_pexp(G)). If it is not abelian, it constructs a list l formed by G. For every element N of l, it takes the maximal subgroups of N which are normal in G and give cyclic quotient. If any of these subgroups is a member of an Iwasawa triple, it is computed with the function <code class="func">IwasawaTripleWithSubgroup</code> (<a href="chap5.html#X812C3A017F534CC0">5.3-3</a>) and the value is returned. If not, N is removed from the l and these maximal subgroups of N are added to l. This follows until an Iwasawa triple is found or the list l is empty. Since normal subgroups with cyclic quotient are contained in a unique maximal chain, no subgroup appears twice in this algorithm.

The algorithm also takes into account the fact that a Iwasawa group of exponent 4 must be abelian or a direct product of a quaternion group of order 8 and an elementary abelian 2-group.

For the trivial group, it returns the triple composed by the trivial group, its identity element, and the prime 3.

<div class="example"><pre>
gap> e:=ExtraspecialGroup(27,3);
<pc group of size 27 with 3 generators>
gap> IwasawaTriple(e);
fail
gap> e:=ExtraspecialGroup(27,9);
<pc group of size 27 with 3 generators>
gap> IwasawaTriple(e);
[ Group([ f1, f3 ]), f2, 1 ]
</pre></div>

<a id="X8121A92F7EABEA0C" name="X8121A92F7EABEA0C"></a>

<h5>5.3-5 IsIwasawaSylow</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIwasawaSylow</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
This function returns <code class="keyw">true</code> if <var class="Arg">G</var> has an Iwasawa (modular) Sylow <var class="Arg">p</var>-subgroup, and <code class="keyw">false</code> otherwise.

Recall that a p-group P has a modular subgroup lattice, or is an Iwasawa group, when all subgroups of P are permutable. It is clear that a p-group has a modular subgroup lattice if and only if it is a T-group.

The implementation of this function begins by searching for a pair of subgroups that do not permute. In this case, the function returns <code class="keyw">false</code>. The maximum number of pairs to be checked here is controlled by the variable <code class="func">PermutMaxTries</code> (<a href="chap3.html#X85CFBEBB83245E3A">3.2-1</a>), which is assigned to 10 by default. If no such pair is found, the algorithm looks for an Iwasawa triple for a Sylow p-subgroup P of G. If there exists one such triple (X,b,s) with s ≥ 2 when p = 2 or the group is a direct product of a quaternion group of order 8 and an elementary abelian 2-group, then it returns <code class="keyw">true</code>; else it returns <code class="keyw">false</code>.

The values of the attributes <code class="func">IsPTGroup</code> (<a href="chap4.html#X7CF2634F84EC572B">4.2-2</a>) and <code class="func">IsTGroup</code> (<a href="chap4.html#X7930E5AD817BE4B3">4.2-1</a>) for P are set by the function.

<div class="example"><pre>
gap> e:=ExtraspecialGroup(27,9);
<pc group of size 27 with 3 generators>
gap> IsIwasawaSylow(e,3);
true
</pre></div>

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