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{\bf BOUNDING NUMBERS AND HEIGHTS OF CHARACTERS IN $p$-CONSTRAINED GROUPS}\\
       \emph{ Geoffrey R. Robinson\\
      School of Mathematics and Statistics\\
       University of Birmingham\\
       Birmingham B15 2TT\\
       England}
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\noindent {\bf INTRODUCTION:} Positive confirmation of  Brauer's
$k(B)$-problem ( which is to prove that the number of ordinary
irreducible characters in a $p$-block $B$ is at most $|D|,$
where $D$ is a defect group for $B$ ) has recently been
achieved in the case that $B$ is a block of a $p$-solvable group,
which represents the culmination of the work of many authors, beginning
with Nagao [6] and Kn\"orr [4].
By contrast, the $k(B)$ problem for $p$-constrained groups
remains open at present.

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In this article, we present a new character-theoretic conjecture for
$p$-constrained groups, which is, for $p$-solvable groups, strictly
stronger than the $k(B)$-problem. The new conjecture seems neither to imply, nor to be implied by,
the $k(B)$-problem for the general $p$-constrained group. In some ways,
though, it is more precise for such a group. A key observation is Theorem 1,
which may be of independent interest, and which clears the way for
the character-theoretic methods successfully employed in the
$k(GV)$-problem to be used in more general situations. In particular,
we prove that the new conjecture holds in fairly general
circumstances, to be described more precisely
later. The generalized characters constructed in Theorem 5 may
also be of independent interest.