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

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Abstract Let H be a subgroup of a finite group G. We say that H satisfies the partial $$ \Pi $$ Π -property in G if there exists a G-chief series $$ \varGamma _{G}: 1 =G_{0}< G_{1}< \cdot \cdot \cdot < G_{n}= G $$ Γ 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 ∩ G i / G i - 1 ) | is a $$ \pi (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) $$ π ( H G i - 1 / G i - 1 ∩ G i / G i - 1 ) -number for every G-chief factor $$ G_{i}/G_{i-1} $$ G i / G i - 1 of $$ \varGamma _{G} $$ Γ G , $$1\le i\le n$$ 1 ≤ i ≤ 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 $$ \Pi $$ Π -property.