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- Jun 20, 2014

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Why must a group with cyclic automorphism group be abelian?

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- Thread starter
- Moderator
- #1

- Jun 20, 2014

- 1,892

-----

Why must a group with cyclic automorphism group be abelian?

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- Jun 20, 2014

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In particular, $\text{Inn}(G)$, the subgroup of inner automorphisms, is cyclic.

But $\text{Inn}(G) \cong G/Z(G)$, where $Z(G)$ is the center of $G$.

If $G/Z(G)$ is cyclic, $G$ is abelian (the following is the standard proof):

Let $xZ(G)$ be the generator of $G/Z(G)$. Then any $g \in G$ is of the form $x^kz$, with $k \in \Bbb Z$, and $z \in Z(G)$.

So given $g,h \in G$, we have:

$gh = (x^kz_1)(x^mz_2) = x^kx^m(z_1z_2) = x^{k+m}z_1z_2 = x^{m+k}z_2z_1 = x^mx^k(z_2z_1) = (x^mz_2)(x^kz_1) = hg$.

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