If $A$, $B$ and $C$ are finite abelian groups that obey the following exact sequence $$A\rightarrow B\rightarrow C\rightarrow1$$ and $$A[m]:=\{a\in A:a^m=1\}$$ is the following inequality true or false, why? $$|B[m]|\leq |A[m]||C[m]|$$
I know that it is false when I take away the finite condition. If it is false, is there some other nice upper bound on $|B[m]|$?
One way to start is by restricting attention to $p$ primary invariant, but I am not sure how to proceed. Maybe taking the quotient of $A[p]$ and $f(A[p])$ will help where $f:A\rightarrow B$ is the homomorphism in the exact sequence, and then induction, but I am not sure if the details follow.
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