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From: weemba@brahms.BERKELEY.EDU (Matthew P. Wiener)
Newsgroups: net.math
Subject: Re: A simple group theory problem
Message-ID: <11641@ucbvax.BERKELEY.EDU>
Date: Fri, 31-Jan-86 20:36:09 EST
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Posted: Fri Jan 31 20:36:09 1986
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In article <14907@rochester.UUCP> sher@rochester.UUCP (David Sher) writes:
>                                           Here is the problem: Is
>every abelian (commutative) group the multiplicative group of some
>number n?

For n>2, -1 != 1 mod n.  But (-1)^2 = 1, so every such group for n>2
contains an element of order 2.  Thus the cyclic group of order 3, for
example, cannot be such a group.

A much harder approach (ie the one I first tried) is to find out what
every such group looks like.  The random facts you quoted are part of
that method.  See T Apostol, _Introduction to Analytic Number Theory_
for a nice low-key discussion.

ucbvax!brahms!weemba	Matthew P Wiener/UCB Math Dept/Berkeley CA 94720