Is it true that a finite group with squarefree order has periodic group cohomology (with trivial coefficients)?


I cannot see why this would be the case, but I'm looking at a paper which seems to implicitly say it's true [Vogel, ON STEENROD'S PROBLEM FOR NON ABELIAN FINITE GROUPS, p.1].



Finite group with squarefree order has periodic cohomology?
All subgroups are then squarefree, and by FTAG all such abelian subgroups are cyclic. Now, a group has periodic cohomology iff all its abelian subgroups are cyclic (Theorem VI.9.5 of Brown's book) -- this is proved by first reducing to Sylow p-subgroups.


