English

Abstract

For a finite group G and a fixed Sylow p-subgroup P of G, Ballester-Bolinches and Guo proved in 2000 that G is p-nilpotent if every element of P ∩ G′ with order p lies in the center of NG(P) and when p = 2, either every element of P ∩ G′ with order 4 lies in the center of NG(P) or P is quaternion-free and NG(P) is 2-nilpotent. Asaad introduced weakly pronormal subgroup of G in 2014 and proved that G is p-nilpotent if every element of P with order p is weakly pronormal in G and when p = 2, every element of P with order 4 is also weakly pronormal in G. These results generalized famous Ito’s Lemma. We are motivated to generalize Ballester-Bolinches and Guo’s Theorem and Asaad’s Theorem. It is proved that if p is the smallest prime dividing the order of a group G and P, a Sylow p-subgroup of G, then G is p-nilpotent if G is S4-free and every subgroup of order p in P ∩ Px ∩ $$G^{\mathfrak{N}_{\mathfrak{p}}}$$ is weakly pronormal in NG(P) for all x ∈ GNG(P), and when p = 2, P is quaternion-free, where $$G^{\mathfrak{N}_{\mathfrak{p}}}$$ is the p-nilpotent residual of G.