Example GrpGPC_Conjugacy (H81E12)

We again consider the nilpotent group G := D₁₆ wreath 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

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>;
> 

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

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

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

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

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

---

[Next][Prev] [Right] [Left] [Up] [Index] [Root]