On homotopy representations with the same dimension function (Q1108626)

\textit{T. tom Dieck} and \textit{T. Petrie} [Homotopy representations of finite groups, Publ. Math. Inst. Hautes Etud. Sci. 56, 129-170 (1982; Zbl 0507.57025)] introduced the notion of a homotopy representation of a finite group G and, using the degree function, they showed that the group of homotopy representations of G with the same dimension function is isomorphic to the Picard group Pic(G). This allows to distinguish between the stable homotopy types of homotopy representations with the same dimension function. \textit{E. Laitinen} [Lect. Notes Math. 1217, 210-248 (1986; Zbl 0628.57019)] obtained the unstable version of the result by showing that two homotopy representations of G with the same dimension function c are G-homotopy equivalent if and only if an invariant in the unstable Picard group \(Pic_ n(G)\) vanishes. The author of the paper studies the number Num(G,n) of the G-homotopy types of homotopy representations of G with the same dimension function n. He shows that \(Num(G,n)\leq | Pic_ n(G)|\) and the equality holds under certain hypotheses fulfilled, for example, when G is a finite nilpotent group of odd order. The author computes the order of \(Pic_ n(G)\) for any finite group G. He also discusses a similar problem for finite homotopy representations of G, and for an abelian group G of odd order, he describes the corresponding number \(Num_ f(G,n)\) by using the Swan homomorphisms. The author concludes with explicit calculations of \(Num_ f(G,n)\) for \(G={\mathbb{Z}}/p\times {\mathbb{Z}}/p,\) p an odd prime.