Set Covering and Serreʹs Theorem on the Cohomology Algebra of a p-Group

We define a group theoretical invariant, denoted by s(G), as a solution of a certain set covering problem and show that it is closely related to chl(G), the cohomology length of a p-group G. By studying s(G) we improve the known upper bounds for the cohomology length of a p-group and determine chl(G) completely for extra-special 2-groups of real type.