arXiv subjects

Zhengtian Qiu

Publications and source records attributed to Zhengtian Qiu.

8 recordsLinked to original sources

arXiv2025

On the $IC$-$Π$-property of subgroups of finite groups

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the $ Π$-property in $ G $ if $ | G/K : N_{G/K}((H \cap L)K/K)| $ is a $ π(( H \cap L)K/K ) $-number for any chief factor $ L/K $ of $ G $; and we call that $ H $ satisfies the $ IC $-$ Π$-property in $ G $ if $ H\cap [H, G] $ satisfies the $ Π$-property in $ G $. In this paper, we obtain a criterion of a normal subgroup being contained in the $ p\mathfrak{U} $-hypercenter of a finite group by the $IC$-$ Π$-property of some $ p $-subgroups.

math.GR↗

arXiv2024

A note on $ p $-supersolubility of finite groups

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies $ \mathscr L $-$ Π$-property in $ G $ if $ | G / K : N _{G / K} (HK/K)| $ is a $ π(HK/K) $-number for all maximal $ G $-invariant subgroup $ K $ of $ H^{G} $. In this paper, we give a characterization of finite $p$-supersoluble groups under assumption that some subgroups of prime power order satisfy $ \mathscr L $-$ Π$-property.

math.GR↗

arXiv2024

On the partial $ \mathscr L $-$ Π$-property of subgroups of finite groups

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \mathscr L $-$ Π$-property in $ G $ if $ H\unlhd G $, or if $ | G / K : \mathrm{N} _{G / K} (HK/K)| $ is a $ π(HK/K) $-number for any $ G $-chief factor of type $ H^{G}/K $ with $ H_{G}\leq K $. In this paper, we investigate the structure of finite groups under the assumption that some subgroups of prime power order satisfy the partial $ \mathscr L $-$ Π$-property.

math.GR↗

arXiv2024

Finite groups with some subgroups of prime power order satisfying the partial $ Π$-property

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

math.GR↗

arXiv2024

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

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}(HK/K\cap L/K )| $ is a $ π(HK/K\cap L/K) $-number. In this paper, 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.

math.GR↗

arXiv2024

On the partial $Π$-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ Π$-property in $ G $ if if there exists a chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that for every $ G $-chief factor $ G_{i}/G_{i-1} $ $ (1\leq i\leq n) $ of $ \varGamma_{G} $, $ | G / G_{i-1} : N_{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| $ is a $ π(HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) $-number. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial $ Π$-property on the structure of a finite group.

math.GR↗

arXiv2024

On the partial $ Π$-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 $ Π$-property in $ G $ if if there exists a chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that for every $ G $-chief factor $ G_{i}/G_{i-1} (1\leq i\leq n) $ of $ \varGamma_{G} $, $ | G / G_{i-1} : N_{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| $ is a $ π(HG_{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 $ Π$-property on the structure of a finite group.

math.GR↗

arXiv2023

Finite groups with some subgroups satisfying the partial $ Π$-property

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ Π$-property in $ G $ if there exists a chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that for every $ G $-chief factor $ G_{i}/G_{i-1} $ $(1\leq i\leq n) $ of $ \varGamma_{G} $, $ | G / G_{i-1} : N _{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})| $ is a $ π(HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1}) $-number. In this paper, we investigate how some subgroups satisfying the partial $Π$-property influence the structure of finite groups.

math.GR↗