On a generalisation of finite $T$-groups

Let $sigma ={sigma_i |iin I}$ 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 $H=H_0leq H_1leq cdots leq H_n=G$ 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.