If $\#A$ and $m$ are relatively prime, then $a\mapsto ma$ is automorphism? Is it true if $A$ is a finite, abelian group and $m$ is some integer relatively prime to the order of $A$, then the map $a\mapsto ma$ is an automorphism?
It's left as an exercise in some course notes, but I cannot verify it.
In particular it is used to prove that Hall complements exist for abelian Hall subgroups of finite groups.

Hint: since $\gcd(m,\#A)=1$, you can find an integer $d$ such that $dm\equiv1\pmod{\#A}$. Show that the map $a \mapsto da$ is the inverse of $a \mapsto ma$.
