[GAP Forum] A conjugacy question
Colva Roney-Dougal
colva at mcs.st-and.ac.uk
Thu Oct 1 10:25:44 BST 2009
Dear Joe
> 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?
No, but they are conjugate in the normaliser of C (in some common
overgroup C). Suppose K^x = H and in particular that k^x = h, and let
c \in C.
Then ck = kc so c^x k^x = k^x c^x. Hence c^x h = h c^x. This holds for
all h \in H, so c^x \in C. Thus x \in N_G(C).
Sadly, computing normalisers is nasty!
Colva
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