The Glauberman correspondence and stable multiplicity-free subgroups. (Q1758434)

Let \(G\) be a finite group, and let \(H\) be a subgroup of \(G\). Let \(\text{Irr}(G,H)\) be the set of pairs \((\chi,\theta)\) where \(\chi\) is an irreducible character of \(G\) and \(\theta\) is an irreducible character of \(H\) and the multiplicity of \(\theta\) in the restriction of \(\chi\) to \(H\) is at least one. Let \(\text{Cl}(G,H)\) be the set of \(H\)-conjugacy classes in \(G\). We say that \(H\) is a multiplicity free subgroup of \(G\) if whenever \(\chi\) is an irreducible character of \(G\) and \(\theta\) is an irreducible character of \(H\), then the multiplicity of \(\theta\) in the restriction of \(\chi\) to \(H\) is at most one. \(\text{Cl}(G,H)\) and \(\text{Irr}(G,H)\) have the same cardinality if and only if \(H\) is a multiplicity-free subgroup. The paper shows that if in addition a solvable group \(A\) of coprime order to \(G\) is acting on \(G\) and stabilizing \(H\), and \(H\) is a multiplicity-free subgroup of \(G\), then \(\text{Cl}(G,H)\) and \(\text{Irr}(G,H)\) are permutation isomorphic as \(A\)-sets. The proof relies on properties of the Glauberman correspondence.