Finite group with periodic cohomology

Definition
A finite group with periodic cohomology is a finite group satisfying the following equivalent conditions:


 * 1) Every abelian subgroup is cyclic
 * 2) Every subgroup whose order is a square of a prime, is cyclic
 * 3) Every Sylow subgroup has rank 1
 * 4) All the Sylow subgroups for odd primes are cyclic, and the 2-Sylow subgroups are either cyclic or generalized quaternion
 * 5) Every subgroup of the group is a Schur-trivial group

The equivalence of definitions depends on the classification of groups of prime power order that have rank 1.

Metaproperties
Any subgroup of a finite group with periodic cohomology again has periodic cohomology. This is most easily seen from the condition that every Abelian subgroup is cyclic.

A quotient of a finite group with periodic cohomology again has periodic cohomology. This is most easily seen from the description in terms of Sylow subgroups.