General Group Properties

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General Group Properties

Subsections

IsAbelian(G) : GrpGPC -> BoolElt

True if the group G is abelian, false otherwise.

IsConsistent(G) : GrpGPC -> BoolElt

True if the stored presentation for G is consistent, false otherwise.

IsCyclic(G) : GrpGPC -> BoolElt

True if the group G is cyclic, false otherwise.

IsElementaryAbelian(G) : GrpGPC -> BoolElt

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

True if the group G is finite, false otherwise.

IsNilpotent(G) : GrpGPC -> BoolElt

True if the group G is nilpotent, false otherwise. This function uses an algorithm described in [Lo98].

IsPerfect(G) : GrpGPC -> BoolElt

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

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

True if the group G is solvable, false otherwise. Every polycyclic group is solvable.

PresentationIsSmall(G) : GrpGPC -> BoolElt

True if only small integers occur in the presentation of G, false otherwise. The category transfer functions FPGroup and PCGroup currently support only groups with a small presentation.

General Properties of Subgroups

IsCentral(G, H) : GrpGPC, GrpGPC -> BoolElt

True if the subgroup H of the group G lies in the centre of G, false otherwise.

IsNormal(G, H) : GrpGPC, GrpGPC -> BoolElt

True if the subgroup H of the group G is a normal subgroup of G, false otherwise.

Properties of Subgroups Requiring a Nil-po-tent Covering Group

The functions described in this section require the existence of a nilpotent covering group. The are based on algorithms published in [Lo98].

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

True if the subgroup H of the nilpotent group G is self-normalising in G, false otherwise.

Example `GrpGPC_SubgroupStructure (H24E9)`

We define a group G on 5 generators a, ..., e of infinite order by fixing the commutators of the generators: (b, a) = e^2, (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

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

We define the subgroups generated by a, ..., 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

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

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

This intersection C is cyclic and central in G.

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

Example `GrpGPC_SubgroupStructure2 (H24E10)`

Consider the nilpotent group G := D_(16) wreath 2 generated by the 5 generators a, b, c, d, t with the relations a^2 = 1, b^(16) = 1, b^a = b^(15) c^2 = 1, d^(16) = 1, d^c = d^(15) t^2 = 1, a^t = c, b^t = d, c^t = a, 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

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 @}

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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