Hecke Operators I'm currently working my way through the section on Hecke operators in Serre's book. In the proof that $ T(m)T(n) = T(mn) $ for all $ m,n \in \mathbb{Z}_{\geq 1} $ with $ \textit{gcd}(m,n) = 1 $.
In this proof Serre makes the claim that a group $ \Gamma/\Gamma'' $ of order $ mn $ decomposes uniquely into a direct sum of a group of order $ m $ and a group of order $ n $. He attributes this to something called $ \textit{Bezout's theorem} $.
I have googled $ \textit{Bezout's theorem} $ and can't seem to find anything that can imply this claim. In particular I can't see why $ \Gamma/\Gamma'' $ cannot be the direct sum  of two groups of orders $ m' $ and $ n' $ such that $ m'n' = mn $. If someone could explain why this cannot be the case, or point me towards the correct theorem I would be most greatful.

A finite abelian group $G$ can be uniquely written as a direct sum of its prime power torsion parts:
$$
G=\bigoplus_pG_p,
$$
where for each prime $p$ we have
$$
G_p=\{x\in G\mid p^mx=0\,\text{for some natural number $m$}\}.
$$
The $m$-part then simply collects those $G_p$ that $p\mid m$. Ditto for the $n$-part.

This is the first time I hear about this being called Bezout's theorem. Frenchmen?
