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.

Textbook references

 * , Page 187, Theorem 3.14, Section 5.3 ($$p'$$-automorphisms of $$p$$-groups)