# Abelian group with order product of primes = cyclic?

1. Jun 13, 2012

### nonequilibrium

It seems rather straight forward that if you have an abelian group G with $\# G = p_1 p_2 \cdots p_n$ (these being different primes), that it is cyclic. The reason being that you have elements $g_1, g_2, \cdots g_n$ with the respective prime order (Cauchy's theorem) and their product will have to have the order of G. Rather simple, but I wanted to check that I'm not overlooking something simple because I find the result rather interesting although I was never told this in any of my algebra classes, which strikes me as strange.

2. Jun 13, 2012

### Barre

I think this can be confirmed by invariant factor decomposition (http://en.wikipedia.org/wiki/Finitely-generated_abelian_group) although really Cauchy's theorem should be sufficient for a proof. $G \cong Z_{k_1} \times Z_{k_2} \times \ldots \times Z_{k_n}$ such that $k_1 \vert k_2 \vert \ldots k_{n-1} \vert k_n$. All $k_i$ are coprime by hypothesis so there can only be one factor.