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From: sher@rochester.UUCP (David Sher)
Newsgroups: net.math
Subject: A simple group theory problem
Message-ID: <14907@rochester.UUCP>
Date: Thu, 30-Jan-86 01:20:42 EST
Article-I.D.: rocheste.14907
Posted: Thu Jan 30 01:20:42 1986
Date-Received: Sat, 1-Feb-86 06:10:38 EST
Distribution: net
Organization: U. of Rochester, CS Dept.
Lines: 20

I have been bothered for some time (so long that I don't remember when
I thought of it) by a relatively simple group theory problem.  I will
try to define it unambiguously.  Consider the operation multiply on 
Z sub n (integers mod n).  Consider the set of integers less than n
relatively prime to n including 1.  This set when combined with the
operation of multiplication mod n is a group with identity 1.  Call
this group the multiplicative group of n.  Here is the problem: Is
every abelian (commutative) group the multiplicative group of some
number n?  Random facts that seem relevant are that the multiplicative
group of every prime p might be (I'm not sure I remember this right)
isomorphic to the cyclic group in p - 1 elements.  Also the
multiplicative group of 8 is not cyclic but generated by two
generators of size (I know I got this word wrong but you understand me
if not work out the group yourself) 2. I accept mail or you can post
if you think your answer is of general interest.

-- 
-David Sher
sher@rochester
seismo!rochester!sher