\title{An algebraic model for chains on $\Omega BG\phat$}
\author{Dave Benson}
\address{Department of Mathematics, University of Aberdeen,
Aberdeen AB24 3UE}
\begin{abstract}
We provide an interpretation of the homology of the loop space
on the $p$-completion of the classifying space of a finite group
in terms of representation theory, and demonstrate how to compute
it. We then give the following reformulation. If $f$ is an idempotent in $kG$
such that $f.kG$ is the projective cover of the trivial module $k$,
and $e=1-f$, then we exhibit isomorphisms for $n\ge 2$:
\begin{align*} 
H_n(\Omega BG\phat;k) &\cong \Tor_{n-1}^{e.kG.e}(kG.e,e.kG) \\
H^n(\Omega BG\phat;k) &\cong \Ext^{n-1}_{e.kG.e}(e.kG,e.kG). 
\end{align*}
Further algebraic structure is examined, such as products and coproducts,
restriction and Steenrod operations.
\end{abstract}