{\bf Involutions, weights and $p$-local structure}\\
\emph{Geoffrey R. Robinson\\
Institute of Mathematics,\\
Fraser Noble Building,\\
University of Aberdeen\\
Aberdeen AB24 3UE,\\
Scotland\\} 


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{\bf Abstract}
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\emph{We prove that for an odd prime $p,$ a finite group $G$ with no
element of order $2p$ has a $p$-block of defect zero if it has a
non-Abelian Sylow $p$-subgroup or more than one conjugacy class of
involutions. For $p=2,$ we prove similar results using elements of order $3$ in place
of involutions. We also illustrate (for an arbitrary prime $p$) that
certain pairs $(Q,y),$ with a $p$-regular element $y$ and $Q$ a
maximal $y$-invariant $p$-subgroup, give rise to $p$-blocks of
defect zero of $N_{G}(Q)/Q$ and we give lower bounds for the number of
such blocks which arise. This relates to the weight conjecture of
J.L. Alperin.}

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\noindent {\bf Keywords:} Block, Involution.\\
{\bf AMS subject classification}: 20C20