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The two element group has a centre (of course it does, it's abelian) but no outer automorphisms. I can come up with a (convoluted) countably infinite non-abelian example. Is there a finite one? If so what is the smallest?

Is there a name for this sort of group? ("Outerless?" Since a group without a centre is "centreless", or "complete" if it has no outermorphisms either.)

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up vote 3 down vote accepted

The smallest example is 2×AΓL(1,8) of order 336. A fairly standard permutation representation is generated by (+,-), (∞,0)(1,3)(2,6)(4,5), (1,2,3,4,5,6,7), and (1,2,4)(3,6,5), or you can use more GAP friendly points to get (9,10), (8,7)(1,3)(2,6)(4,5), (1,2,3,4,5,6,7), and (1,2,4)(3,6,5). AΓL(1,8) is a complete, solvable group with no normal subgroup of index 2.

I just checked the groups ordered by size, skipping nilpotent groups since they always have outer automorphisms. Probably one could be a bit smarter, looking for complete groups with no 2-quotients, but I wasn't sure the smallest example would be of the form 2×Complete.

There are examples like 2.Sz(8) which have no outer automorphisms, have a non-trivial center, but where the quotient by the center is not complete (so it is definitely not 2×Complete).

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Brilliant. Thanks! –  anon Sep 9 '10 at 22:14
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$C_2\times M_{11}$.

Steve

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Oh, good. Yes, that works. Is it the smallest? –  anon Sep 8 '10 at 23:30
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