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.

中文翻译：

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具有一些弱前正定子群的有限群的 $p$-nilpotency

对于有限群 G 和 G 的固定 Sylow p-子群 P，Ballester-Bolinches 和 Guo 在 2000 年证明，如果 P ∩ G' 的每个元素都位于 NG(P) 的中心，则 G 是 p 幂零的当 p = 2 时，P ∩ G' 的每个 4 阶元素都位于 NG(P) 的中心，或者 P 是无四元数且 NG(P) 是 2 幂零的。Asaad 于 2014 年引入了 G 的弱顺式子群，并证明了如果 P 中 p 阶的每个元素在 G 中都弱顺式，并且当 p = 2 时，P 中的每个 4 阶元素也在 G 中弱顺式，则 G 是 p 幂零的。这些结果概括了著名的伊藤引理。我们有动力推广 Ballester-Bolinches 和郭的定理以及阿萨德的定理。证明如果 p 是除群 G 和 P 的阶的最小素数，则是 G 的 Sylow p-子群，