In article <61bpmr$sv$1@nntp.Stanford.EDU>, hwatheod@leland.Stanford.EDU (theodore hwa) wrote: > > Einar Andreas R|dland (firstname.lastname@example.org) 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. email@example.com 2 dificult 2 work out." http://www.maths.ex.ac.uk/~rjc/rjc.html Iain M. Banks - Feersum Endjinn
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