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

Abstract

Abstract Let Irr \u2061 ( G ) {\\operatorname{Irr}(G)} denote the set of complex irreducible characters of a finite group G, and let cd \u2061 ( G ) {\\operatorname{cd}(G)} be the set of degrees of the members of Irr \u2061 ( G ) {\\operatorname{Irr}(G)} . For positive integers k and l, we say that the finite group G has the property 𝒫 k l {\\mathcal{P}^{l}_{k}} if, for any distinct degrees a 1 , a 2 , … , a k ∈ cd \u2061 ( G ) {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 \u2061 ( a 1 , a 2 , … , a k ) {\\gcd(a_{1},a_{2},\\dots,a_{k})} is at most l - 1 {l-1} . In this paper, we classify all finite almost simple groups satisfying the property 𝒫 3 2 {\\mathcal{P}_{3}^{2}} . As a consequence of our classification, we show that if G is an almost simple group satisfying 𝒫 3 2 {\\mathcal{P}_{3}^{2}} , then | cd \u2061 ( G ) | ⩽ 8 {\\lvert\\operatorname{cd}(G)\\rvert\\leqslant 8} .