Identify conjugacy class of coset in factor group in GAP

Question

I wish to use GAP to identify the conjugacy class of a given element $[g]$ in the factor group $G/H$ where $H$ is a finite-index normal subgroup of a finitely presented group $G$. I know how to compute a list of right coset representatives of $H$:

T:=RightTransversal(G,H);

and can determine to which coset $[g]$ a given element $g\in G$ belongs using PositionCanonical(T,g).

To study the conjugacy classes of $G/H$, I construct the factor group and compute its conjugacy classes:

quot:=FactorGroup(G,H);
CC:=ConjugacyClasses(quot);

To check whether $[g]$ belongs to, say, the third conjugacy class of $G/H$, I thought of doing:

IsConjugate(quot,RightCoset(H,g),Representative(CC[3]));

but GAP returns an error message of the form:

Error, no 1st choice method found for `IsConjugate' on 3 arguments at /usr/local/gap-4.11.1/lib/methsel2.g:249 called from
IsConjugate( quot, RightCoset(...

which I've truncated at the end. Any suggestions on how to do this computation properly?

Answer 1

The issue is that GAP represents (for efficiency reasons) the factor group in a new way, i.e. its elements are not cosets (but e.g. permutations). The key is to use the natural homomorphism:

nat:=NaturalHomomorphismByNormalSubgroup(G,H);
quot:=Image(nat,G);
CC:=ConjugacyClasses(quot);

and then either (analog to your command)

IsConjugate(quot,Image(nat,g),Representative(CC[3]));

or (shorter to type and potentially faster:

Image(nat,g) in CC[3]