LAUSR

Abstract Let Irr⁡(G){\operatorname{Irr}(G)} denote the set of complex irreducible characters of a finite group G, and let cd⁡(G){\operatorname{cd}(G)} be the set of degrees of the members of Irr⁡(G){\operatorname{Irr}(G)}. For positive integers k and l, we say that the finite group G has the property 𝒫kl{\mathcal{P}^{l}_{k}} if, for any distinct degrees a1,a2,…,ak∈cd⁡(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⁡(a1,a2,…,ak){\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 𝒫32{\mathcal{P}_{3}^{2}}. As a consequence of our classification, we show that if G is an almost simple group satisfying 𝒫32{\mathcal{P}_{3}^{2}}, then |cd⁡(G)|⩽8{\lvert\operatorname{cd}(G)\rvert\leqslant 8}.