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Is it true that every finite group of even order $>2$ has an automorphism of order $2?$

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up vote 3 down vote accepted

This is not true. For each odd prime $p,$ there are finite $p$-groups whose automorphism group is a $p$-group (in fact, "most" $p$-groups have that property in an appropriate sense- see work of U. Martin and coauthors). Take the direct product of such a $p$-group with a cyclic group of order $2$ and the automorphism group of the resulting group is still a $p$-group.

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I'm not familiar with the mentioned name. Could you be more precise which articles/books you're referring to? – Ehsan M. Kermani Dec 25 '11 at 4:51
To find exampless of $p$-groups whose automorphism group is a $p$-group is not hard. I think there are example in Gorenstein's text "Finite Groups" (Harper and Row, 1968). This is all you really need. The fact that almost all $p$-groups have automorphism group a $p$-group is what Ursula Martin and coathors proved, but that is not essential to answer your question, it was for general interest. – Geoff Robinson Dec 25 '11 at 18:56

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