On the $p$-length of a finite factorizable group with given permutability conditions for subgroups of factors

A subgroup $A$ of a group $G$ is called $tcc$-subgroup in $G$, if there is a subgroup $T$ of $G$ such that $G=AT$ and for any $X\leq A$ and for any $Y\leq T$ there exists an element $u\in\langle X, Y\rangle$ such that $XY^u\leqslant G$. Suppose that $G=AB$ is a product of two $p$-soluble $tcc$-subgroups $A$ and $B$. We give a bound of the $p$-length of $G$ from the nilpotent class and the number of generators of $A_p$ and $B_p$, where $A_p$ and $B_p$ are the Sylow subgroups of $A$ and $B$ respectively.