Multiplicities of irreducible components of restrictions of complex representations of finite groups to certain subgroups

Abstract

Let G be a finite group, n an integer and P a field; we say that G ∈ Rn(P) if for any irreducible representation X of G there exists a proper subgroup H (depending on X) and an irreducible representation ξ of H which is a component of X|H with multiplicity ≤ n. We prove that G ∈ Rn(P) for an algebraically closed field P provided S ∈ Rn(P) for the universal central extensions S of all nonabelian simple quotients of G. We also prove that the universal central extensions of all projective special linear groups PSL(m.,q), of alternating groups Am and Suzuki groups Sz(q) belong to R1(C).