Cyclic GroupsDate: 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/ |
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