GAP QA #9
All conjugate Subgroups
Alexander Hulpke (
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
or look at particular transversal elements
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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