Confirming a conjecture by Alperin in an unpublished manuscript on finding a structural explanation as to why the McKay conjecture in group respresentation theory should be true, the authors prove the following result: If \(G\) is a \(p\)-solvable finite group with a self-normalizing Sylow \(p\)-subgroup \(P\), then for every \(z\in G\) the quantity \(\sum_{\chi\in\text{Irr}_{p'}(G)}\sum_{g\in P'zP'}\chi(g)\) is a rational integer divisible by \(|P|\). This remains no longer true if the \(p\)-solvability hypothesis is dropped, as \(S_5\) with \(p=2\) shows. Motivated by Alperin’s and Broué’s work, they also present a generalized version of their result indicating that for \(p\)-solvable groups with self-normalizing Sylow \(p\)-subgroups, double cosets of the form \(P'zP'\) can replace elements in Broué’s perfect isometries.