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