# All conjugate Subgroups

Keywords: Subgroups, Transversal

## Respondent:

Alexander Hulpke ( hulpke@math.colostate.edu)

## Question:

I have got a group G and a subgroup H and would like to compute all the
conjugate subgroups H^g = g^-1*H*g of H in G. Further, for every conjugate
subgroup of H in G, call it C, I would like to know one g satisfying C=H^g.

You want to use the right transversal. If \$H\$ is a subgroup, let
T:= RightTransversal( G, Normalizer(G,H) );
The elements of \$T\$ are in one-to-one correspondence to the subgroups
conjugates to \$H\$.
You can run through the transversal in a loop:
for i in T do
rep:=i;
conj:=H^i;
...
od;
or look at particular transversal elements
T[i]
Finally you can identify the tranversal position of a coset by
`PositionCanonical' (gives the position of the representative for the same
coset. This might not be the same representative).
The transversal object uses some magic to avoid writing out a list of all
elements at once, so this approach is also memory efficient.