Structure of blocks with normal defect and abelian inertial quotient

Abstract

Let k be an algebraically closed field of prime characteristic p. Let $kGe$ be a block of a group algebra of a finite group G, with normal defect group P and abelian $p'$ inertial quotient L. Then we show that $kGe$ is a matrix algebra over a quantised version of the group algebra of a semidirect product of P with a certain subgroup of L. To do this, we first examine the associated graded algebra, using a Jennings–Quillen style theorem. As an example, we calculate the associated graded of the basic algebra of the nonprincipal block in the case of a semidirect product of an extraspecial p-group P of exponent p and order $p^3$ with a quaternion group of order eight with the centre acting trivially. In the case of $p=3$ , we give explicit generators and relations for the basic algebra as a quantised version of $kP$ . As a second example, we give explicit generators and relations in the case of a group of shape $2^{1+4}:3^{1+2}$ in characteristic two.