On the $p$-length of a product of two $B$-groups

A finite non-nilpotent group is called a $B$-group if every proper subgroup of its quotient group by Frattini subgroup is primary. The $p$-length $l_p(G)$ of a finite $p$-soluble group, which is the product of two $B$-subgroups, is studied. It has been proved that $l_p(G)\leqslant 1$ if $p$ does not divide the index of one of the $B$-subgroups.