On a new type of products of finite groups

Let $G_{1}$ and $G_{2}$ be subgroups of a finite group $G$. We say that $G_{1}$ and $G_{2}$ are totally semipermutable if $G_{1}G_{2}$ is a subgroup of $G$, and whenever $P$ is a Sylow subgroup of $G_{1}$ and $Q$ is a Sylow subgroup of $G_{2}$ with $(|P|,|Q|)=1$, we have every subgroup of $P$ permutes with every subgroup of $Q$. In this paper, we study the influence of totally semipermutable subgroups on the structure of finite groups. Our results improve and extend some results in the literature.