Let G be a finite group and k a field of characteristic \(p>0\). If M is a finitely generated kG-module and \(...\to P_ m\to P_{m-1}\to...\to P_ 0\to M\to 0\) a minimal projective resolution of M, then the complexity, \(c_ G(M)\), of M is the least integer \(s\geq 0\) such that \(\lim_{m\to \infty}\dim_ kP_ m/m^ s=0.\) \textit{J. L. Alperin} and \textit{L. Evens} [in J. Pure Appl. Algebra 22, 1-9 (1981; Zbl 0469.20008)] proved that \(c_ G(M)=\max_{E}(c_ E(M_ E))\) as E runs over the elementary abelian p-subgroups of G, where \(M_ E\) is the kE-module which is the restriction of the kG-module to E. In this beautiful paper the author gives a method for computing the complexity. Let E be an elementary abelian p-group with basis \(\{a_ 1,...,a_ n\}\) and \(\phi\) a linear automorphism of \(k(a_ 1- 1)\oplus...\oplus k(a_ n-1).\) Then \(\phi\) has a unique extension \({\bar \phi}\) to a ring isomorphism of kE. A k-generalized subgroup \(E_ 0\) of E is a subgroup of the multiplicative group \({\bar \phi}\)(E). We can now state the main theorems: Thm. 1. Let M be a kE-module of complexity d. The there exists a k- generalized subgroup \(E_ 0\) of order \(p^{n-d}\) such that \(c_{E_ 0}(M_{E_ 0})=0\) (hence \(M_{E_ 0}\) is projective). - Thm. 2. \(c_ E(M)=\nu_ p(| E|)-\max_{E_ 0} \nu_ p(| E_ 0|)\) as \(E_ 0\) runs over the k-generalized subgroups with \(M_{E_ 0}\) projective. \((\nu_ p\) is the usual p-adic valuation, \(| E_ 0|\) the order of \(E_ 0.)\)- Thm. 3. Let \(c_ G(M)=d.\) Then \(p^{m- d}| \dim_ k M,\) where \(p^ m=\min_{E}(| E|)\) as E runs over the maximal elementary abelian p-groups. Moreover, there is a plethora of very interesting results and new proofs of well-known theorems in the area. One of the main points of the proof is the study of the relation between the cohomology of a group G and the cohomology of a normal subgroup of G of index p. \textit{J. Carlson} has later obtained a new proof of Thm. 1 [in J. Algebra 85, 104-143 (1983; Zbl 0526.20040)].