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Show that no group can have its automorphism group cyclic of odd order.

I have shown it only if $G$ is cyclic, but I could not do that if $G$ is not cyclic. Can you help?

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    http://www.physicsforums.com/showthread.php?t=1731482011-05-19
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    Note that this is not quite true: a group of order $1$ or $2$ has trivial automorphism group, which is cyclic of odd order! But the proof sketched below accounts for this...2011-05-19
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    you always have inversion2011-05-19
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    @yoyo: Inversion is an anti-automorphism.2011-05-19
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    I think @Peter L. Clark is right. I see the same question on Page 30 of Derek J.S. Robinson's A Course in the Theory of Groups (GTM 80), with ">1" added to the end.2011-08-03

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