Solve Group Theory Problem - Prime Order of G must be p^n

[equivalence]

Hi

I have a problem I just can't seem to solve, even though the solution shouldn't be too hard

Let G be a finite abelian group and let p be a prime.
Suppose that any non-trivial element g in G has order p. Show that the order of G must be p^n for some positive integer n.

Anyone got any ideas about how to approach this??

thanks,

 

[matticus]

Suppose there is another prime q that divides the order of the group and show there must be an element of order q.

 

[equivalence]

but is it the case that for all factors of the order of a group there is an element of that order?? i am soo confused..

 

[mathmadx]

You know Lagranges theorem..? Consider the subgroup generated by g,- what's his order?. Well, if you like carefully at what " generates" means, youll see that the order of the subgroup generated by g is also p.