Almost simple groups with no product of two primes dividing three character degrees

Abstract

Let {\operatorname{Irr}(G)} denote the set of complex irreducible characters of a finite group G, and let {\operatorname{cd}(G)} be the set of degrees of the members of {\operatorname{Irr}(G)} . For positive integers k and l, we say that the finite group G has the property {\mathcal{P}^{l}_{k}} if, for any distinct degrees {a_{1},a_{2},\dots,a_{k}\in\operatorname{cd}(G)} , the total number of (not necessarily different) prime divisors of the greatest common divisor {\gcd(a_{1},a_{2},\dots,a_{k})} is at most {l-1} . In this paper, we classify all finite almost simple groups satisfying the property {\mathcal{P}_{3}^{2}} . As a consequence of our classification, we show that if G is an almost simple group satisfying {\mathcal{P}_{3}^{2}} , then {\lvert\operatorname{cd}(G)\rvert\leqslant 8} .