Re: Centralizers, Normalizers and the Center
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Thu, 7 Jul 2005 16:59:43 +0000 (UTC)
In article <1120682296.196741.91150@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<themadhatter012@xxxxxxxxx> wrote:
>If H is a subgroup of G is it true that
>
>Z(G) is a subgroup of C_H(G) which is a subgroup of N_H(G)?
>
>I believe that it is.
>
By definition:
Z(G) = {g in G : gx=xg for all x in G}
I don't know what you mean by C_H(G). It usually means
{h in H : hg = gh for all g in G}
but it seems to me that you want it to mean
{g in G : gh = hg for all h in H}.
If so, this is usually denoted by C_G(H), not C_H(G).
Same with N_H(G). I would usually interpret this as
{ h in H : hGh^{-1} = G}
but it seems to me that you want it to be
{ g in G : gHg^{-1} = H}
which I normally see written as N_G(H), not as N_H(G).
So, assuming the definitions are:
Z(G) = {g in G : gx=xg for all x in G}
C_H(G) = {g in G : gh = hg for all h in H}.
N_H(G) = { g in G : gHg^{-1} = H}
then the answer is: yes, Z(G) is contained in C_H(G) is contained in
N_H(G). Now prove it.
>If H is just a subset of G the above is not necessarily true, right?
No. If H is a subset, the inclusions still hold. Think about it.
>Are there any easy examples of the above 2 cases?
What two cases?
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx
.
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