\begin{center} {\bf On
Brauer's $k(B)$-problem for blocks of $p$-solvable groups
with non-Abelian defect groups}\\
\emph{Geoffrey R. Robinson,\\
School of Mathematics and Statistics,\\
University of Birmingham,\\
Birmingham B15 2TT\\
}



\end{center}

\medskip
\noindent {\bf INTRODUCTION:} The proof of Brauer's
$k(B)$-problem for $p$-solvable groups has recently been completed
by Gluck,Magaard, Riese and Schmid [2]. Their latest work
settled the only outstanding case ( $p = 5$) and represents the
completion of the work of a series of authors dating back
over 40 years ( see, for example, earlier work by various combinations
of these authors for small primes and papers such as
[1],[3],[5],[6],[7]).

\medskip
Brauer's $k(B)$-problem is to prove that if the
$p$-block $B$ has defect group $D,$ then $k(B)\leq |D|,$
where $k(B)$ denotes the number of ordinary irreducible characters
of $B$. Two of the earliest published reductions of this problem
(for $B$ a $p$-block of a $p$-solvable group) were achieved by H. Nagao,
who showed that it was sufficient to prove that when $H$ is a
$p$-solvable group with $O_{p'}(H) = 1,$
then $k(H) \leq |H|_{p},$ and that furthermore, it was sufficient
to consider the case that $H = GV$ for $G$ a $p^{\prime}$-group
acting faithfully and irreducibly on the elementary Abelian
$p$-group $V.$ We will prove here that the inequality of
Brauer's $k(B)$ problem is strict when $B$ is a $p$-block of
a $p$-solvable group with a non-Abelian defect group
$D.$ Our proof makes use of
the affirmative answer to the $k(B)$-problem
for $p$-solvable groups and does not provide an alternative proof
of that result. Our main result is:

\medskip
\noindent {\bf THEOREM 1 :} \emph{ Let $B$ be a $p$-block of a
$p$-solvable group which has a non-Abelian defect group $D.$
Then $k(B) < |D|.$}