Abelian group with order product of primes = cyclic?

[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.

 

[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.