General Group Properties#

IsAbelian(G): GrpGPC -> BoolElt#: Returns true if the group \(G\) is abelian, false otherwise.

IsCyclic(G): GrpGPC -> BoolElt#: Returns true if the group \(G\) is cyclic, false otherwise.

IsElementaryAbelian(G): GrpGPC -> BoolElt#

Returns true if the group \(G\) is elementary abelian, false otherwise. The following definition is used:

A group \(G\) is called elementary abelian if it is an abelian \(p\)-group of exponent \(p\) for some prime \(p\).

IsFinite(G): GrpGPC -> BoolElt#: Returns true if the group \(G\) is finite, false otherwise.

IsNilpotent(G): GrpGPC -> BoolElt#: Returns true if the group \(G\) is nilpotent, false otherwise. This function uses an algorithm described in [Lo, 1998].

IsPerfect(G): GrpGPC -> BoolElt#: Returns true if the group \(G\) is perfect, false otherwise. A polycyclic group \(G\) is perfect, if and only if it is trivial.

IsSimple(G): GrpGPC -> BoolElt#: Returns true if the group \(G\) is simple, false otherwise. A polycyclic group is simple, if and only if it is cyclic of prime order.

IsSoluble(G): GrpGPC -> BoolElt#
IsSolvable(G): GrpGPC -> BoolElt#: Returns true if the group \(G\) is solvable, false otherwise. Every polycyclic group is solvable.

General Properties of Subgroups#

IsCentral(G, H): GrpGPC, GrpGPC -> BoolElt#: Returns true if the subgroup \(H\) of the group \(G\) lies in the centre of \(G\), false otherwise.

IsNormal(G, H): GrpGPC, GrpGPC -> BoolElt#: Returns true if the subgroup \(H\) of the group \(G\) is a normal subgroup of \(G\), false otherwise.

Properties of Subgroups Requiring a Nilpotent Covering Group#

The functions described in this section require the existence of a nilpotent covering group. The are based on algorithms published in [Lo, 1998].

IsConjugate(G, H, K): GrpGPC, GrpGPC, GrpGPC -> BoolElt, GrpGPCElt#: Given groups \(G\), \(H\) and \(K\) with a nilpotent common covering group, return the value true if there exists \(c\in G\) such that \(H^c = K\). If so, the function returns such a conjugating element as second value.

IsSelfNormalising(G, H): GrpGPC, GrpGPC -> BoolElt#
IsSelfNormalizing(G, H): GrpGPC, GrpGPC -> BoolElt#: Returns true if the subgroup \(H\) of the nilpotent group \(G\) is self-normalising in \(G\), false otherwise.

Example: Subgroup Structure (ex-905fd9)#

We define a group \(G\) on 5 generators \(a,\ldots,e\) of infinite order by fixing the commutators of the generators:

\[(b,a) = e^2, \quad (d,c)=e^3\]

All other pairs of generators commute.

> F<a,b,c,d,e> := FreeGroup(5);
> rels := [ b^a = b*e^2, b^(a^-1) = b*e^-2, d^c = d*e^3,
>           d^(c^-1) = d*e^-3 ];
> G<a,b,c,d,e> := quo< GrpGPC: F | rels >;
> IsNilpotent(G);
true

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Since \(G\) is nilpotent, we can compute intersections of subgroups of \(G\).

We define the subgroups generated by \(a,\ldots,e\) and their nontrivial commutator groups as subgroups of \(G\).

> H1 := sub<G|a>;
> H2 := sub<G|b>;
> H3 := sub<G|c>;
> H4 := sub<G|d>;
> H5 := sub<G|e>;
>
> C12 := CommutatorSubgroup(H1, H2);
> {@ G!x : x in PCGenerators(C12) @};
{@ e^2 @}
> C12 subset H5;
true
>
> C34 := CommutatorSubgroup(H3, H4);
> {@ G!x : x in PCGenerators(C34) @};
{@ e^3 @}
> C34 subset H5;
true

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Finally, we compute the intersection \(C\) of \(C12\) and \(C13\).

> C := C12 meet C34;
> {@ G!x : x in PCGenerators(C) @};
{@ e^6 @}

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This intersection \(C\) is cyclic and central in \(G\).

> IsCyclic(C);
true
> IsCentral(G, C);
true

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Example: Subgroup Structure2 (ex-bc5c0c)#

Consider the nilpotent group \(G := D_{16}\wr 2\) generated by the 5 generators \(a,b,c,d,t\) with the relations

\[a^2 = 1,\quad b^{16} = 1,\quad b^a = b^{15}\]

\[c^2 = 1,\quad d^{16} = 1,\quad d^c = d^{15}\]

\[t^2 = 1,\quad a^t = c,\quad b^t = d,\quad c^t = a,\quad d^t = b\]

(All other pairs of generators commute.)

> F<t, a,b, c,d> := FreeGroup(5);
> G<t, a,b, c,d> := quo<GrpGPC: F | a^2, b^16, b^a=b^15,
>                                   c^2, d^16, d^c=d^15,
>                      t^2, a^t=c, b^t=d, c^t=a, d^t=b>;
> IsNilpotent(G);
true

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Since \(G\) is nilpotent, we can compute normalisers and centralisers in \(G\).

We define the (dihedral) subgroup \(D3\) of \(G\) generated by \(ac\) and \(bd\) and compute its normaliser in \(G\) and its centraliser in the (dihedral) subgroup \(D2\) of \(G\) generated by \(c\) and \(d\).

> D2 := sub<G|c,d>;
>
> D3<u,v> := sub<G|a*c, b*d>;
> D3;
GrpGPC : D3 of order 2^5 on 2 PC-generators
PC-Relations:
    u^2 = Id(D3),
    v^16 = Id(D3),
    v^u = v^15
>
> N3 := Normaliser(G, D3);
> PCGenerators(N3, G);
{@ t, a * c, b * d, d^8 @}
>
> C3 := Centraliser(D2, D3);
> PCGenerators(C3, G);
{@ d^8 @}

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Finally we compute the centraliser of the element \(t\) in \(G\).

> Ct := Centraliser(G, t);
> PCGenerators(Ct, G);
{@ t, a * c, b * d @}

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