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 ,
Mojtaba Ghirati 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