FINITE GROUPS WITH GIVEN WEAKLY $\tau_{\sigma}$-QUASINORMAL SUBGROUPS

Let $\sigma=\{{\sigma_i|i\in I}\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group.  A set $\mathcal{H} $ of  subgroups  of $G$ is said to be a \textit{complete Hall $\sigma$-set} of $G$ if every non-identity member of $\mathcal{H}$ is a Hall  $\sigma_i$-subgroup of $G$ for some $i\in I$ and $\mathcal{H}$ contains exactly one Hall $\sigma_i$-subgroup of $G$ for every $i$ such that $\sigma_i\cap \pi(G)\neq \emptyset$.  Let $\tau_{\mathcal{H}}(A)=\{ \sigma_{i}\in \sigma(G)\backslash \sigma(A) \ |\  \sigma(A) \cap \sigma(H^{G})\neq \emptyset$ for a Hall $\sigma_{i}$-subgroup $H\in \mathcal{H}$$\}$.  A subgroup $A$ of $G$ is said to be \textit{$\tau_{\sigma}$-permutable} or \textit{$\tau_{\sigma}$-quasinormal} in $G$ with respect to  $\mathcal{H}$ if $AH^{x}=H^{x}A$ for all $x\in G$ and $H\in \mathcal{H}$ such that $\sigma(H)\subseteq \tau_{\mathcal{H}}(A)$, and \textit{$\tau_{\sigma}$-permutable} or \textit{$\tau_{\sigma}$-quasinormal} in $G$ if $A$  is \textit{$\tau_{\sigma}$-permutable} in $G$ with respect to some complete Hall $\sigma$-set of $G$. We say that a subgroup $A$ of $G$ is  \textit{weakly $\tau_{\sigma}$-quasinormal} in $G$ if $G$ has a  $\sigma$-subnormal subgroup $T$ such that $AT=G$ and $A\cap T\leq A_{\tau_{\sigma}G}$, where  $A_{\tau_{\sigma}G}$ is the subgroup generated by all those subgroups of $A$ which are $\tau_{\sigma}$-quasinormal in $G$. We study the structure of $G$ being based on the assumption that some subgroups of $G$ are weakly $\tau_{\sigma}$-quasinormal in $G$.