Math Is Fun Forum

sam123

Guest

group theory proof

Let G be a finite cyclic group of order n, and let g E G such that |g| = n. Prove that the elsements 1,g,g^2,...,g^n-1 are distinct. Deduce that g is a generator for G.

(where E means g is an element of G!)

Getting stuck with this one, any help would be greatly appreciated, cheers!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: group theory proof

Hi sam123

I think you could assume g^i = g^j where i and j are not equal.  Say i > j.  Times g^i by enough 'g's to make it into the identity and the other by the same.  You've got a contradiction as you now have an power of 'g' below 'n' that is the identity.  So no two are the same.

That means you've got 'n' distinct elements so you got them all => g has generated them all.

Bob

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Children are not defined by school ...........The Fonz
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Sometimes I deliberately make mistakes, just to test you!  …………….Bob