Blocks with normal abelian defect and abelian p′ inertial quotient

Let k be an algebraically closed field of characteristic p⁠, and let O be either k or its ring of Witt vectors W(k)⁠. Let G be a finite group and B a block of OG with normal abelian defect group and abelian p′ inertial quotient L⁠. We show that B is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan’s conjecture. For O=k⁠, we give an explicit description of the basic algebra of B as a quiver with relations. It is a quantized version of the group algebra of the semidirect product P⋊L⁠