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

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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