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Finite groups whose all proper subgroups are 𝒞 -groups

Pengfei Guo;  Jianjun Liu — 2018

Czechoslovak Mathematical Journal

A group G is said to be a 𝒞 -group if for every divisor d of the order of G , there exists a subgroup H of G of order d such that H is normal or abnormal in G . We give a complete classification of those groups which are not 𝒞 -groups but all of whose proper subgroups are 𝒞 -groups.

Finite groups with some SS-supplemented subgroups

Mengling Jiang;  Jianjun Liu — 2021

Czechoslovak Mathematical Journal

A subgroup H of a finite group G is said to be SS-supplemented in G if there exists a subgroup K of G such that G = H K and H ∩ K is S-quasinormal in K . We analyze how certain properties of SS-supplemented subgroups influence the structure of finite groups. Our results improve and generalize several recent results.

The p -nilpotency of finite groups with some weakly pronormal subgroups

Jianjun Liu;  Jian Chang;  Guiyun Chen — 2020

Czechoslovak Mathematical Journal

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 N G ( P ) and when p = 2 , either every element of P ∩ G ' with order 4 lies in the center of N G ( P ) or P is quaternion-free and N G ( 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...

A note on the Π -property of some subgroups of finite groups

Zhengtian Qiu;  Guiyun Chen;  Jianjun Liu — 2024

Czechoslovak Mathematical Journal

Let H be a subgroup of a finite group G . We say that H satisfies the Π -property in G if for any chief factor L / K of G , | G / K : N G / K ( H K / K ∩ L / K ) | is a π ( H K / K ∩ L / K ) -number. We obtain some criteria for the p -supersolubility or p -nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the Π -property.

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