SOME CRITERIA FOR SOLVABILITY AND NILPOTENCY OF FINITE GROUPS BY STRONGLY MONOLITHIC CHARACTERS

Gagola and Lewis ['A character theoretic condition characterizing nilpotent groups', Comm. Algebra27 (1999), 1053-1056] proved that a finite group G is nilpotent if and only if chi(1)(2) divides vertical bar G:ker chi vertical bar for every irreducible character chi of G. The theorem was later generalised by using monolithic characters. We generalise the theorem further considering only strongly monolithic characters. We also give some criteria for solvability and nilpotency of finite groups by their strongly monolithic characters.