Proving o(G) Divisible by p in Abelian Groups

[Singularity]

Hey guys. I've been stuck on the following thing for a little while now. Some help would be appreciated.

If p divides o(G) (G an abelian group and p a prime), then show that
G(p) = {g from G | o(g) = p^k for some k }

I keep going round in circles.

P.S. - this is not a homework question, just something I saw in an abstract algebra book that they stated without proof.

 

[d_leet]

Hey guys. I've been stuck on the following thing for a little while now. Some help would be appreciated.

If p divides o(G) (G an abelian group and p a prime), then show that
G(p) = {g from G | o(g) = p^k for some k }

I keep going round in circles.

P.S. - this is not a homework question, just something I saw in an abstract algebra book that they stated without proof.

What is G(p)? Are you sure this wasn't a definition?

 

[robert Ihnot]

What about Cauchy's Theorem?

 

[matt grime]

If p divides o(G) (G an abelian group and p a prime), then show that
G(p) = {g from G | o(g) = p^k for some k }

If you don't define G(p) for us we can't help. (The sylow p subgroup perhaps? but that is an easy exercise - all elements of order p^k lie in some sylow subgroup, and there is only one in an abelian group - so it must be something more difficult than that.)

 

[Singularity]

Did it:)

Thanks to all who posted. My apologies for not defining G(p) properly. In any event, I solved the problem. The fundamental theorem of arithmetic did the trick (I underestimated the strength of the uniqueness of prime factorization in my earlier attempts!). Ciao