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Order of abelian subgroup of the automorphism group of an abelian group

Claim) Suppose $A$ is a finite non-trivial Abelian group. Let $B$ be an Abelian subgroup of ${\rm{Aut}}(A)$. Then, $|B|\leq |A|-1$ if $|A|$ and $|B|$ are coprime. An elementary proof by induction that ...
KhashF's user avatar
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Order of abelian subgroup of the automorphism group of an abelian group

As Nick Gill points out, one can certainly have $|B| > |A|$ if you do not assume coprimality. If you assume coprimality then yes $|B| < |A|$. If $A_p$ is the Sylow $p$-subgroup of $A$ then $\...
Sean Eberhard's user avatar

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