\begin{center}
{\bf ON THE FOCAL DEFECT GROUP OF A BLOCK, CHARACTERS OF HEIGHT ZERO, AND LOWER DEFECT GROUP MULTIPLICITIES}\\
\emph{Geoffrey R. Robinson,\\
Department of Mathematics,\\
University of Aberdeen,\\
Scotland}\\
April 2008
\end{center}

\medskip
\noindent {\bf ABSTRACT:} We discuss the focal subgroup of the
defect group $D$ of a $p$-block $B,$ which we refer to as the
\emph{focal defect group}, and denote by $D_{0}.$ We note that (the
character group) of $D/D_{0}$ acts ( in a defect (or height)
preserving fashion) on irreducible characters in $B,$ and prove that
the action on irreducible characters of height zero is semi-regular.
We also prove  that all orbits under this action have length
divisible by $[Z(D): D_{0} \cap Z(D)].$ As applications, we prove
that all Cartan invariants for $B$ are divisible by $[Z(D): D_{0}
\cap Z(D)],$ that if ${\rm Out}(D)$ is a $p$-group ( and $D \neq
1),$ then the number of irreducible characters of height zero in $B$
is divisible by $p$ and that if $Z(D) \not \leq D_{0},$ then the
block $B$ is of Lefschetz type (see [5]).