On the cohomology of finite groups and the applications to modular representations (Q809177)

In [J. Pure Appl. Algebra 22, 43-56 (1981; Zbl 0483.20003)] \textit{J. F. Carlson} introduced a certain condition on \(p\)-groups to study the structure of periodic modules. Here the author extends Carlson's condition to arbitrary finite groups and obtains similar results. In particular, if \(G\) is a finite group and \(k\) a field of prime characteristic then it is shown that there is a number \(n(G)\) so that the period of a periodic \(kG\)-module divides \(2n(G)\). Moreover, it is shown that a \(kG\)-module is projective if and only if \(Ext_{kG}^{2n(G)}(A,A)=0\). A similar result to the latter has also been obtained by \textit{P. W. Donovan} [in J. Algebra 117, 434-436 (1988; Zbl 0647.20010)] for certain finite groups.