On a Generalisation of Finite T-Groups

Let $\sigma =\{\sigma_i |i\in I\}$ is some partition of all primes $\mathbb{P}$ and $G$ a finite group. A subgroup $H$ of $G$ is said to be $\sigma$-subnormal in $G$ if there exists a subgroup chain $H=H_0\leq H_1\leq \cdots \leq H_n=G$ such that either $H_{i-1}$ is normal in $H_i$ or $H_i/(H_{i-1})_{H_i}$ is a finite $\sigma_j$-group for some $j \in I$ for $i = 1, \ldots, n$. We call a finite group $G$ a $T_{\sigma}$-group if every $\sigma$-subnormal subgroup is normal in $G$. In this paper, we analyse the structure of the $T_{\sigma}$-groups and give some characterisations of the $T_{\sigma}$-groups.

[1]  W. Guo,et al.  Groups with maximal subgroups of Sylow subgroups σ-permutably embedded , 2016 .

[2]  A. Skiba.  On σ-subnormal and σ-permutable subgroups of finite groups , 2015 .

[3]  Klaus Doerk,et al.  Finite Soluble Groups , 1992 .

[4]  W. Guo,et al.  ON THE LATTICE OF -SUBNORMAL SUBGROUPS OF A FINITE GROUP , 2017 .

[5]  W. Gaschütz.  Gruppen, in denen das Normalteilersein transitiv ist. , 1957 .

[6]  Xianbiao Wei.  On weakly m-σ-permutable subgroups of finite groups , 2019, Communications in Algebra.

[7]  A. Skiba.  A generalization of a Hall theorem , 2016 .

[8]  D. Robinson.  A Course in the Theory of Groups , 1982 .

[9]  A. Ballester-Bolinches,et al.  Products of Finite Groups , 2010 .

[10]  W. Guo,et al.  On σ-supersoluble groups and one generalization of CLT-groups , 2018, Journal of Algebra.

[11]  P. Hall,et al.  Theorems Like Sylow's , 1956 .

[12]  A. Skiba,et al.  On finite groups with σ-subnormal Schmidt subgroups , 2017 .

[13]  W. Guo,et al.  On weakly $sigma$-permutable subgroups of finite groups , 2016, 1608.03224.

[14]  P. Hermann.  Onπ-quasinormal subgroups in finite groups , 1989 .

[15]  A. Skiba.  Some characterizations of finite σ -soluble PσT -groups , 2018 .

[16]  Finite groups with permutable complete Wielandt sets of subgroups , 2015 .

[17]  Finite groups with pro-normal subgroups , 1969 .

[18]  W. Guo,et al.  Finite groups whose n-maximal subgroups are σ-subnormal , 2016, 1608.03353.

[19]  On τσ-quasinormal subgroups of finite groups , 2017 .

[20]  A. Skiba.  On Some Results in the Theory of Finite Partially Soluble Groups , 2016 .

[21]  On sublattices of the subgroup lattice defined by formation Fitting sets , 2020 .

[22]  W. Guo.  Structure Theory for Canonical Classes of Finite Groups , 2015 .