# Structure theorem for fixed point-free automorphism group of p-group

## Statement

Suppose $P$ is a group of prime power order, i.e., a finite $p$-group for some prime number $p$. Suppose $G \le \operatorname{Aut}(P)$ is a group all of whose non-identity elements are fixed point-free automorphisms: none of them fixes any non-identity element of $P$. Then, the following are true:

1. The order of $G$ is relatively prime to $p$.
2. Every Abelian subgroup of $G$ is cyclic.
3. If $q$ and $r$ are primes dividing the order of $G$, every subgroup of $G$ of order $qr$ is cyclic.

Note that (2) tells us that $G$ is a finite group with periodic cohomology, and, combined with the classification of finite p-groups of rank one, we get:

• All the $q$-Sylow subgroups of $G$ for odd primes $p$ are cyclic.
• The $2$-Sylow subgroups of $G$ are either cyclic or generalized quaternion.

(3), however, places stronger restrictions on $G$ than simply being a finite group with periodic cohomology.

## References

### Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, More info, Page 187, Theorem 3.14, Section 5.3 ($p'$-automorphisms of $p$-groups)