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Topic: Example
Replies: 11   Last Post: Oct 8, 1997 7:28 AM

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Robin Chapman

Posts: 695
Registered: 12/6/04
Re: When every group of order n is abelian (Was: example)
Posted: Oct 7, 1997 4:34 AM
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In article <61bpmr$sv$1@nntp.Stanford.EDU>,
hwatheod@leland.Stanford.EDU (theodore hwa) wrote:
> Einar Andreas R|dland ( wrote:
> :
> : Also, if o(G)=p^2, G is abelian.
> :
> This is slightly off the subject of the original thread but it brings up
> the question of when every group of order n is abelian. We know that
> 1) There's a non-abelian group of order p^3 for every prime p:
> the group of 3 by 3 upper triangular matrices having 1's on the main
> diagonal over the field Z_p.
> 2) If p and q are primes such that p divides q-1 then there's a
> non-abelian semidirect product of Z_p by Z_q.

3) Suppose p and q are primes with p dividing q^2 - 1. The elementary
abelian group Z_q x Z_q has an automorphism of order q^2 - 1, so also
one of order p, and this can be used to construct a non-abelian
semidirect product of Z_p with Z_q x Z_q.

> Therefore certainly 2 necessary conditions for n are that it be cube-free
> and for every two primes p,q dividing n, p cannot divide q-1. Is this
> sufficient?
> A _Monthly_ article from a few years ago (June-July 1992 ?! I know it was
> a summer issue) proved that if "cube-free" is replaced by "square-free"
> then the above is sufficient for every group of order n to be cyclic. (The
> condition was stated as gcd(n,phi(n))=1, which is the same thing.)

The three conditions above are the only restricitions. I.e., every group
of order n is abelian if and only if
(i) n is cubefree
(ii) If q is a prime factor of n, then no other prime factor of n divides
q - 1.
(ii) If q is a prime factor of n and q^2 divides n, then no other prime
factor of n divides q^2 - 1.

This problem has come up from time to time on sci.math. If I recall
correctly a proof is outlined in the exercises of Dummit & Foote's book
on Abstract Algebra, but I don't have my copy at hand.

Robin Chapman "256 256 256.
Department of Mathematics O hel, ol rite; 256; whot's
University of Exeter, EX4 4QE, UK 12 tyms 256? Bugird if I no. 2 dificult 2 work out." Iain M. Banks - Feersum Endjinn

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