Extra special defect groups of order [formula omitted] and exponent p

Abstract

In this paper we give results on ( p-)blocks with the defect groups isomorphic to an extra special group of order p 3 . We are particularly interested in the number of irreducible ordinary characters, and the number of irreducible Brauer characters in the block. The situation splits naturally into two cases according to the exponent of the extra special group in this paper we concentrate on the exponent p case. We prove that if p > 7 , then there exists a non-major subsection. We give some conditions which guarantee that Brauer's k ( B ) -Conjecture is satisfied.