\author{David Benson, Radha Kessar, and Markus Linckelmann} 


\address{David Benson \\ 
Institute of Mathematics\\ 
Fraser Noble Building\\
University of Aberdeen\\ 
King's College\\ 
Aberdeen AB24 3UE\\ 
United Kingdom}

\address{Radha Kessar and Markus Linckelmann \\
School of Mathematics, Computer Science \& Engineering \\
Department of Mathematics \\
City, University of London \\
Northampton Square \\
London EC1V 0HB \\
United Kingdom}

\subjclass[2010]{20C20, 20J06}

\keywords{Finite groups, block theory,  abelian defect, Frobenius twist}

\title{Blocks with normal abelian defect and abelian $p'$ inertial quotient}

\begin{document}

\begin{abstract}
Let $k$ be an algebraically closed field of characteristic $p$,
and let $\CO$ be either $k$ or its ring of Witt vectors $W(k)$.
Let $G$ a finite group and $B$ a block of $\CO G$ with normal
abelian defect group and abelian $p'$ inertial quotient.
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 $\CO=k$, we give an explicit description of the basic algebra
of $B$ as a quiver with relations. It is a quantised version
of the group algebra of the semidirect product $P\rtimes L$.

\end{abstract}