From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Group Theory
Date: 1 Oct 1997 13:41:58 GMT

In article <fkumybe8ad92@forum.swarthmore.edu>,
Mojtaba Ghirati <Ghirati@math.sci.shirazu.ac.ir> wrote:
>We know the center of a finite P_group has at least 2 members. Is it
>true for infinite P_groups? 

No; that's why the proof for finite p-groups must use that counting
argument using the class equation.

Let  U_n  denote the subgroup of  GL(n,p)  consisting of matrices with
0's below and 1's on the main diagonal; it's a Sylow p-subgroup of
GL(n,p). We can view  U_n  as a subgroup of  U_(n+1)  by appending zeros
to the right and bottom of each matrix in  U_n.  Let  U  be the
direct limit of the  U_n 's. (You can think of  U  as the set of 
upper-triangular infinite matrices which differ from the identity in only
finitely-many entries.)  Clearly  U  is a p-group, since every element
in it lies in some  U_i  and hence has order a power of  p. But
it has a trivial center, since every element other than the identity
fails to commute with at least one other matrix: if  M  lies in  U_n,
then it does not commute with this element of  U_(2n):
	I I
	0 I
(Here of course we identify  M  with the element
	M 0
	0 I
of  U_(2n).)

dave