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




Re: When every group of order n is abelian (Was: example)
Posted:
Oct 7, 1997 4:34 AM


In article <61bpmr$sv$1@nntp.Stanford.EDU>, hwatheod@leland.Stanford.EDU (theodore hwa) wrote: > > Einar Andreas Rdland (einara@ulrik.uio.no) 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 nonabelian 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 q1 then there's a > nonabelian 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 nonabelian semidirect product of Z_p with Z_q x Z_q.
> Therefore certainly 2 necessary conditions for n are that it be cubefree > and for every two primes p,q dividing n, p cannot divide q1. Is this > sufficient? > > A _Monthly_ article from a few years ago (JuneJuly 1992 ?! I know it was > a summer issue) proved that if "cubefree" is replaced by "squarefree" > 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. rjc@maths.exeter.ac.uk 2 dificult 2 work out." http://www.maths.ex.ac.uk/~rjc/rjc.html Iain M. Banks  Feersum Endjinn
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