Structure theorem for fixed point-free automorphism group of p-group
From Groupprops
Statement
Suppose is a group of prime power order, i.e., a finite -group for some prime number . Suppose is a group all of whose non-identity elements are fixed point-free automorphisms: none of them fixes any non-identity element of . Then, the following are true:
- The order of is relatively prime to .
- Every Abelian subgroup of is cyclic.
- If and are primes dividing the order of , every subgroup of of order is cyclic.
Note that (2) tells us that is a finite group with periodic cohomology, and, combined with the classification of finite p-groups of rank one, we get:
- All the -Sylow subgroups of for odd primes are cyclic.
- The -Sylow subgroups of are either cyclic or generalized quaternion.
(3), however, places stronger restrictions on 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 (-automorphisms of -groups)