From: IN%"math-fun@mailman.xmission.com" "math-fun" 23-SEP-2003 12:28:46.98 Received: from clyde.concordia.ca (clyde.Concordia.CA [132.205.1.1]) by vax2.concordia.ca (PMDF V6.2 #30759) with ESMTP id <01L105NGXJ4Q006D64@vax2.concordia.ca> for mckay@vax2.concordia.ca (ORCPT mckay@vax2.concordia.ca); Tue, 23 Sep 2003 12:28:46 -0400 (EDT) Received: from mailman.xmission.com (mailman.xmission.com [198.60.22.29]) by clyde.concordia.ca (8.12.10/8.12.10) with ESMTP id h8NGSak9449440 for <mckay@vax2.concordia.ca>; Tue, 23 Sep 2003 12:28:36 -0400 (EDT) Received: from localhost ([127.0.0.1] helo=mailman.xmission.com) by mailman.xmission.com with esmtp (Exim 3.35 #1 (Debian)) id 1A1q1b-0003se-03; Tue, 23 Sep 2003 10:28:27 -0600 Received: from mail.math.princeton.edu ([128.112.18.14]) by mailman.xmission.com with esmtp (Exim 3.35 #1 (Debian)) id 1A1q1Z-0003sL-00 for <math-fun@mailman.xmission.com>; Tue, 23 Sep 2003 10:28:25 -0600 Received: from localhost.localdomain (fine318b.math.princeton.edu [128.112.16.36])h8NGRrNs017817verify=NOT); Tue, 23 Sep 2003 12:27:53 -0400 Received: from fine318b.math.Princeton.EDU (localhost.localdomain [127.0.0.1]) h8NGRrnS007185; Tue, 23 Sep 2003 12:27:53 -0400 Received: from localhost (conway@localhost)h8NGRrfr007181; Tue, 23 Sep 2003 12:27:53 -0400 Date: Tue, 23 Sep 2003 12:27:53 -0400 (EDT) From: John Conway <conway@Math.Princeton.EDU> Subject: Re: [math-fun] non-square products of squares? In-reply-to: <Pine.LNX.4.44.0309231039460.5413-100000@fine318b.math.Princeton.EDU> Sender: math-fun-bounces@mailman.xmission.com To: math-fun <math-fun@mailman.xmission.com> Cc: Marc LeBrun <mlb@fxpt.com> Errors-to: math-fun-bounces@mailman.xmission.com Reply-to: math-fun <math-fun@mailman.xmission.com> Message-id: <Pine.LNX.4.44.0309231209450.6960-100000@fine318b.math.Princeton.EDU> MIME-version: 1.0 Content-type: TEXT/PLAIN; charset=US-ASCII Precedence: list X-BeenThere: math-fun@mailman.xmission.com X-Mailman-Version: 2.1.1 X-Scanned-By: MIMEDefang 2.35 List-Post: <mailto:math-fun@mailman.xmission.com> List-Subscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=subscribe> List-Unsubscribe: <http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun>, <mailto:math-fun-request@mailman.xmission.com?subject=unsubscribe> List-Archive: <http://mailman.xmission.com/cgi-bin/mailman/private/math-fun> List-Help: <mailto:math-fun-request@mailman.xmission.com?subject=help> List-Id: math-fun <math-fun.mailman.xmission.com> Original-recipient: rfc822;mckay@vax2.concordia.ca On Tue, 23 Sep 2003, John Conway wrote:

On Mon, 22 Sep 2003, Dan Hoey wrote:

...

I found two such groups of order 16, but I haven't figured out what their names are. A presentation for the first is

<a,b : a^2 = b^4 = (b a b)^2 = (b^-1 a b a)^2 = 1> .

Its squares are 1, b^2, and (a b)^2, and the product of the last two is not a square.

I don't immediately recognise this group in that presentation. I'll play with it offline to see which one it is.

I've now done so. As I expected, it's what I think of as "the last group" of order 16, which Coxeter calls (4,4|2,2) (except that I may have the wrong style of bracket or punctuation): 1 = a^4 = b^4 = (ab)^2 = (a^-1.b)^2. [This is more symmetrical than the presentation you give - my a can be taken as your ab.] The elements can be written a^i.b^j, and exactly half of them (those for which i+j is odd) have order 4. They are of two "sexes" according as i is odd and j even or i even and j odd, and any two elements of opposite sexes generate the group in the above presentation. The square of an element of order 4 is a^2 or b^2, depending only on its sex, so of course it's an example. [I remark that a^2 and b^2 generate the center.]

...

The other group may be presented <a,b : a^4 = b^4 = a b a^-1 b = 1>.

Its squares are 1, a^2, b^2, and again the product of the two non-identity squares is not a square.

This group is called 4:4 in my system. [In general, A:B denotes the split extension of a group of structure A by one of structure B, and it is usually assumed that it's not the direct product, since we'd call that A x B.]

This group is much easier to think about, since it's that split extension. The square of a^i.b^j is a^(2i) if j is even, and b^2 if j is odd, so superficially it looks very like the other (in the new presentation) in respect of this behavior, though this time there's no symmetry between a and b. When I guessed there wouldn't be examples of order 16, I planned to check (4,4|2,2) just in case, because it's the most interesting one, but forgot. [I "learned" the groups of order 16 by heart long ago, though now I may be a bit shaky. The standard reference for them is a table in the back of Coxeter and Moser, but they describe them in ways that don't tell you the structure, so it's better to use ATLAS-type notations that do.] John Conway _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun