[GAP Forum] A conjugacy question

[Joe Bohanon]

I'm working on a project that requires me to test subgroups of Co1 for 
conjugacy.  The enormous permrep makes the standard backtrack algorithm 
get very bogged down, so I've had to write several routines to speed 
things up.

One place where I don't get very good speed-up is for groups with 
trivial solvable radical.  I had an idea that I'm not sure if it's 
mathematically sound.  Say that I have two groups H and K.  Let C be the 
centralizer of H and D the centralizer of K.  Suppose I can find an 
element conjugating D to C and I conjugate K by it so that H and K are 
in the centralizer of C.  Is it true that if H and K are conjugate in G, 
then they are also conjugate in the centralizer of C?

Thanks
Joe