On the Partial \( \Pi \)-Property of Some Subgroups of Prime Power Order of Finite Groups

Let H be a subgroup of a finite group G. We say that H satisfies the partial \( \Pi \) $\mathrm{\Pi }$ -property in G if there exists a chief series \( \varGamma _{G}: 1 =G_{0}< G_{1}< \cdots < G_{n}= G \) ${\Gamma }_G:1=G_0<G_1<\cdots <G_n=G$ of G such that for every G-chief factor \( G_{i}/G_{i-1} (1\le i\le n) \) $G_i/G_{i-1}(1\le i\le n)$ of \( \varGamma _{G} ,\) ${\Gamma }_G,$ \( | G / G_{i-1}: N _{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| \) $|G/G_{i-1}:N_{G/G_{i-1}}(H{G}_{i-1}/G_{i-1}\cap G_i/G_{i-1})|$ is a \( \pi (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) \) $\pi (H{G}_{i-1}/G_{i-1}\cap G_i/G_{i-1})$ -number. In this paper, we study the influence of some subgroups of prime power order satisfying the partial \( \Pi \) $\mathrm{\Pi }$ -property on the structure of a finite group.