#### GAP QA #9

# 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.
## Answer:

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.

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