A NOTE ON GROUPS WHOSE ALL NON-LINEAR IRREDUCIBLE CHARACTERS ARE DEFECT ZERO

Abstract

Let G be a finite group, χ a non-linear irreducible character of G.  dz(χ) denotes the set of all prime divisors p of |G| such that χ is a p−defect zero character, dz(G) denotes the union of dz(χ) for all non-linear irreducible characters χ of G, i.e., χ ∈ Irr(G|G′).  A finite group G such that \(\bigcap_{χ∈Irr(G|G′)\) dz(χ) ≠ ∅ was studied in 1996.  Finite groups G satisfying dz(χ) = π(χ(1)) for all χ ∈ Irr(G|G′) were classified in 2007.  We are motivated to study more general case, i.e., a finite group G satisfying dz(χ) ≠ ∅ for every χ ∈ Irr(G|G′).  At first, we study a solvable group G and come to a necessary and sufficient condition.  Secondly, for a non-solvable group, we prove that K₃-simple groups can be uniquely determined by dz(G) and the order of G.