2-blocks with an abelian defect group and a freely acting cyclic inertial quotient

Abstract

We study blocks with an abelian defect group and a cyclic inertial quotient acting freely but not transitively. We prove that when p=2, such blocks are inertial, i.e. basic Morita equivalent to their Brauer correspondent. Together with a result of the second author on Singer cycle actions on homocyclic defect groups, this completes the classification of 2-blocks with a cyclic inertial quotient acting freely on an abelian defect group.