Automorphism group of an abelian group abelian

Question

Is the automorphism group of an abelian group abelian? If not is it true if either is finite? I cannot come up with a proof or an example.

score 1 · Accepted Answer · answered Nov 07 '19 at 15:42

1

No, the automorphism group of $C_2\times C_2$ is isomorphic to the non-abelian group $S_3$.

answered Nov 07 '19 at 15:42

Dietrich Burde

Thanks for your quick reply! What is $C_2$? – Pink Flying Elephant Nov 07 '19 at 15:43
$C_2$ is the cyclic group with $2$ elements. – Dietrich Burde Nov 07 '19 at 15:43
$C_n=\mathbb Z/n\mathbb Z=\mathbb Z_n$. – Rushabh Mehta Nov 07 '19 at 15:44
Ah, ok thanks a bunch! – Pink Flying Elephant Nov 07 '19 at 15:44

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