# Does every $p$-group of odd order admit fixed point free automorphisms?

Does every $p$-group of odd order admit fixed point free automorphisms?

equivalently,

Given an odd order $p$-group $P$, is there a group $C$ such that we can form a Frobenius group $P\rtimes C$?

Note that this is not true for $p$-groups of even order, for example $Q_8$ and $C_4$. But I cannot think of an example with odd order.

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Extraspecial groups of order $p^3$ admit no fixed point free automorphisms. – Derek Holt Sep 24 '13 at 22:04
To give an explicit example of Derek's comment: the extraspecial group of order $27$ (the one with all elements of order $3$) has a center of size $3$. Clearly the only prime $q$ which could act fpf on this center is $q=2$. But any group with an fpf automorphism of order $2$ is abelian, contradiction. To carry this further, if a $p$-group $G$ admits an fpf automorphism of prime order $q$, one can bound its nilpotency class in terms of $q$. Thus for any $p$, we can build $p$-groups of maximal class and large order which admit no fpf automorphism. – user641 Sep 24 '13 at 22:18
I think the extraspecial of order 125 and exponent 5 does have a fpf of order 4. – Jack Schmidt Sep 25 '13 at 15:42
@JackSchmidt: I just checked with GAP, I didn't find any. – user641 Sep 25 '13 at 16:08

The following family of $p$-groups provides counter examples $$G = \langle a, b, c,d |a^{p^n}=b^{p^4}=c^{p^4}=d^{p^2}=1, [a,b]=[a,c]=b^{p^2}, [a,d]=c^{p^2}, [b,c]=a^{p^{n-2}}, [b,d]=c^{p^2}, [c,d]=c^{p^2} \rangle$$ with $p$ odd, and $n>3$.
For such a group $G$, every automorphism is central, that is $\operatorname{Aut}(G)$ acts trivially on $G/\operatorname{Z}(G)$. It is easy to see that the number of central automorphisms in that case (actually, for any finite group with no abelian direct factor) is equal to the order of $\operatorname{Hom}\left(G/G',\operatorname{Z}(G)\right)$. Thus $\operatorname{Aut}(G)$ is a $p$-group, so there is no automorphism acting fixed point freely on $G$.
As mentioned by Steve D, it is proved by U. Martin and G. Helleloid that (in some sense) almost finite $p$-groups have an automorphism group of $p$-power order, thus 'almost' $p$-groups have no fixed point free automorphisms.