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 G is a finite group and k is a field of characteristic p, then any kG-module is a direct summand of a module which has a filtration whose sections are induced from elementary abelian p-subgroups of G. We also prove two new theorems which are closely related to Carlson’s theorem.