Conjugacy

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

IsConjugate(G, g, h): GrpGPC, GrpGPCElt, GrpGPCElt -> BoolElt, GrpGPCElt

Given elements \(g\) and \(h\) and a group \(G\), which are contained in some nilpotent common group, return the value true if there exists \(c\in G\) such that \(g^c = h\). If so, the function returns such a conjugating element as second value.

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.

Example: Conjugacy (ex-f79f1e)

We again consider the nilpotent group \(G := D_{16}\wr 2\).

> 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, a test for conjugacy in \(G\) is available.

We define the following subgroups of \(G\): \(D1\) generated by \(a\) and \(b\), \(D2\) generated by \(c\) and \(d\) and \(D3\) generated by \(ac\) and \(bd\).

> D1 := sub<G|a,b>;
> D2 := sub<G|c,d>;
> D3<u,v> := sub<G|a*c, b*d>;
>

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\(D1\) and \(D2\) are, of course, conjugate in \(G\); \(t\) is a conjugating element.

> IsConjugate(G, D1, D2);
true t

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The elements \(b\) and \(d^{-1}\) are conjugate in \(G\); we compute a conjugating element.

> IsConjugate(G, b, d^-1);
true t * a * c

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However, neither the subgroups \(D1\) and \(D2\) nor the elements \(b\) and \(d^{-1}\), are conjugate in the subgroup \(D3\).

> IsConjugate(D3, D1, D2);
false
> IsConjugate(D3, b, d^-1);
false

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