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Let $G$
be a finite group with cyclic Sylow $p$
-subgroups, and let $k$
be a field of characteristic $p$
. Then $H^{*}(BG;k)$
and $H_*(\Omega BG{{}^{{}^{\wedge }}_p};k)$
are $A_\infty$
algebras whose structure we determine up to quasi-isomorphism.
