From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2672 Path: news.gmane.org!not-for-mail From: "Ronald Brown" Newsgroups: gmane.science.mathematics.categories Subject: Re: Extensions of Z+Z by Z Date: Wed, 28 Apr 2004 07:27:54 +0100 Message-ID: References: <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="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018820 5256 80.91.229.2 (29 Apr 2009 15:27:00 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:27:00 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Wed Apr 28 13:58:40 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 28 Apr 2004 13:58:40 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BIsMH-00079o-00 for categories-list@mta.ca; Wed, 28 Apr 2004 13:56:29 -0300 X-Priority: 3 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 52 Original-Lines: 62 Xref: news.gmane.org gmane.science.mathematics.categories:2672 Archived-At: Try the Heisenberg group of upper triangular matrices with 1's on the diagonal and the integers i,j,k in the upper non diagonal entries. 1 k i 0 1 j 0 0 1 This should give the case c=1, but your formula is not quite correct as the RHS does not involve q. Should it be q + j + c.(kr)? Ronnie Brown http://www.bangor.ac.uk/~mas010 ----- Original Message ----- From: "Colin McLarty" To: Sent: Monday, April 26, 2004 3:58 AM Subject: categories: Extensions of Z+Z by Z > 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 > > >