Re: simple proof for a group theory question?
From: Derek Holt (mareg_at_warwick.ac.uk)
Date: 09/27/04
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Date: Mon, 27 Sep 2004 16:00:08 +0000 (UTC)
marian <marian@fasttelco.com> wrote in message news:<cj7muc$kh8$1@news.ks.uiuc.edu>...
> Did anybody see this little result before? The proof I have involves
> actions of some sort and I would like to see a more elementary proof.
>
>
> Let G be a finite group with center Z and for x in G let Cl(x) denote
> the conjugacy class of x in G. Then |xZ\cap Cl(x)| divides both |Z|
> and |Cl(x)|.
I would not necessarily consider a proof involving actions to be
non-elementary. Surely, any study of conjugacy classes must involve
actions?
Let C be the inverse image in G of C_{G/Z}(xZ). Then, for g in G,
g^-1 x g lies in xZ if and only if g is in C. So xZ\cap Cl(x) is the
class of x in C. Since this has size |C/C_C(x)|, and C_C(x) = C_G(x),
we have
|xZ\cap Cl(x)| = |C/C_C(x)| = |C/C_G(x)| divides |G/C_G(x)| =
|Cl(x)|.
Also if, for g in C, we write g^-1 x g = x z_g with z_g in Z, then
we get
Z_g z_h = z_{gh}, so the z_g form a subgroup of Z, and hence |xZ\cap
Cl(x)| divides |Z|.
Derek Holt.
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