Finite Groups with Some Subgroups of Prime Power Order Satisfying the Partial \( \Pi \)-property

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 G-chief series \( \varGamma _{G}: 1 =G_{0}< G_{1}< \cdot \cdot \cdot < G_{n}= G \) ${\Gamma }_G:1=G_0<G_1<···<G_n=G$ of G such that \( | G / G_{i-1}: N _{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| \) $|G/G_{i-1}:N_{G/G_{i-1}}(H{G}_{i-1}/G_{i-1}\cap G_i/G_{i-1})|$ 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 for every G-chief factor \( G_{i}/G_{i-1} \) $G_i/G_{i-1}$ of \( \varGamma _{G} \) ${\Gamma }_G$ , \(1\le i\le n\) $1\le i\le n$ . In this paper, we investigate the structure of a finite group G under the assumption that some subgroups of prime power order satisfy the partial \( \Pi \) $\mathrm{\Pi }$ -property.