Cyclic Groups

Date: 04/18/2002 at 14:26:21
From: Robert
Subject: Cyclic Groups

Prove that a group of order 5 is cyclic.

I have been working on this problem for a while and I can't seem to 
get anywhere on it.  Please help.

Date: 04/18/2002 at 15:51:22
From: Doctor Paul
Subject: Re: Cyclic Groups

This follows from Lagrange's theorem. Pick any nonidentity element in 
the group and consider the subgroup generated by the powers (or 
multiples if the group is additive) of this element. By Lagrange's 
Theorem, the order of this subgroup must divide the order of the 
group.  The only numbers that divide 5 are 5 and 1. The subgroup 
doesn't have order one since we started with a nonidentity element.  
Thus it must have order five and is hence cyclic.  A similar argument 
shows that every group of prime order is cyclic.

I hope this helps.  Please write back if you'd like to talk about 
this some more.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/