Congruences for the Burnside module (Q1409290)

The object under consideration in this paper was defined by \textit{R. Oliver} and \textit{T. Petrie} in [Math. Z. 179, 11-42 (1982; Zbl 0484.57022)]. It is denoted in this paper by \(\Omega(G, \Pi)\), where \(G\) is a finite group, \(\Pi\) a partially ordered set and \(G\) acts on \(\Pi\) as a group of order preserving automorphisms. The author terms \(\Omega(G,\Pi)\) the Burnside module and it consists of equivalence classes of \(\Pi\)-complexes. The precise definition of \(\Omega(G,\Pi)\) and its abelian group structure is beyond the scope of this review and is given in the paper. Motivated by a congruence that holds in the Burnside ring \(\Omega(G)\), the author proves two theorems about congruences that hold in the Burnside module.