From - Thu Sep  3 20:54:37 1998
From: mwdaly@pobox.com (Matthew Daly)
Newsgroups: sci.math
Subject: Probably a simple group theory question
Date: Thu, 03 Sep 1998 17:28:11 GMT
Organization: Eastman Kodak Company
Message-ID: <35eed051.178484500@news.kodak.com>

After many many years, I'm going through my old abstract algebra
textbooks, and I came across a problem that is driving me up a tree.
Here it is:

Show that a group in which every a satisfies a^2 = 1 is abelian.  What
if a^3 = 1 for every a?

The first part is easy; you can show in a single line that both ab and
ba are inverses of ab, so ab=ba for all a and b.  But the second part
has me completely stumped; I can't come up with a proof or a
counterexample, and I'm not even sure which one seems more intuitive to
me.  Any pushes in the correct direction would greatly enhance my
sanity.

-Matthew
--
Matthew Daly       mwdaly@pobox.com     http://www.frontiernet.net/~mwdaly/
My opinions may have changed, but not the fact that I am right - Ashleigh Brilliant
The views expressed here are not necessarily those of my employer, of course.
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There is a non-abelian group of order 27 such that 26 of its elements have 
period 3.  It is not to hard to work out more details from this.
Good luck!

--
John McKay
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Analysing groups with a^3 = 1 for all a is more difficult than for
a^2 = 1, but is still reasonably easy. It turns out that such groups are
nilpotent of class at most 3 (so they are not necessarily abelian).
The largest group B(r,3) with r generators has order  3^m(r), where
m(r) = (r choose 3) + (r choose 2) + r so, for example, m(3)=7.
One reference for the proof is the chapter on the Burnside Problem in
M. Hall, 'The Theory of Groups'.

This is of course a special case of the well-known Burnside Problem.
Let B(r,n) be the 'largest' group with r generators such that a^n=1 for all
elements a. (This is well-defined even when B(r,n) is infinite.)
Then B(r,n) is known to be finite for n=2,3,4 and 6, and infinite for
large enough n - about n>= 8000 is the best relaible result, although
n > 117 has been claimed for odd n.

--
mareg@crocus.csv.warwick.ac.uk (Dr D F Holt)
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Study a group of upper triangular 3 by 3 matrices with entries from
a field of characteristic three (so that 3=0 in the field). Use binomial
expansion of (I+A) to the third power with A strictly upper triangular... 
More hints?

--
Jyrki Lahtonen, Ph.D.
Department of Mathematics,
University of Turku,
FIN-20014 Turku, Finland
From: Jyrki.Lahtonen@utu.fi (Jyrki Lahtonen)