A theorem of Jon F. Carlson on filtrations of modules

We give an alternative proof to a theorem of Carlson [J.F. Carlson, Cohomology and induction from elementary abelian subgroups, Quart. J. Math. 51 (2000) 169–181] which states that if GG is a finite group and kk is a field of characteristic pp, then any kGkG-module is a direct summand of a module which has a filtration whose sections are induced from elementary abelian pp-subgroups of GG. We also prove two new theorems which are closely related to Carlson’s theorem.