[GAP Forum] A conjugacy question

[Colva Roney-Dougal]

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