From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2669 Path: news.gmane.org!not-for-mail From: Colin McLarty Newsgroups: gmane.science.mathematics.categories Subject: Extensions of Z+Z by Z Date: Sun, 25 Apr 2004 22:58:28 -0400 Message-ID: <5.2.0.9.0.20040425221849.01aaaeb8@pop.cwru.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed X-Trace: ger.gmane.org 1241018818 5239 80.91.229.2 (29 Apr 2009 15:26:58 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:26:58 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Apr 27 16:54:05 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 27 Apr 2004 16:54:05 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BIYbu-00070h-00 for categories-list@mta.ca; Tue, 27 Apr 2004 16:51:18 -0300 X-Sender: cxm7@pop.cwru.edu (Unverified) X-Mailer: QUALCOMM Windows Eudora Version 5.2.0.9 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 49 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:2669 Archived-At: After calculating the group extensions of Z+Z by Z, with constant action, I am curious whether the groups have any more natural form than I found. I mean extension of Z+Z by Z in this sense, as a sequence of groups where E need not be commutative: 0 --> Z --> E --> Z+Z --> 0 and the kernel is in the center of E. The form I found is parametrized by the integers this way: For any integer c, the group E_c has triples of integers (i,,j,k) as elements and the multiplication rule is coordinate-wise addition plus an extra bit in the first coordinate. (i,,j,k).(q,r,s) = ( (i+j+c.(kr)), j+r, k+s) When c=0 this is commutative and is just the coproduct Z+Z+Z. In any group E_c, the element (c,0,0) is the commutator of (0,0,1) and (0,1,0). The Baer sum of extensions corresponds to addition of the parameters c as integers. So I understand the group of extensions. Of course I understood it before I calculated it, since it is the second cohomology group of the torus. That is why I tried the algebraic calculation. But is there a natural way to think about each group E_c, for non-zero values of c? Do these groups appear in any other natural way? thanks, colin