On Weakly Factorizable Groups

Groups with complemented subgroups, which are also called completely factorizable groups, were studied by P. Hall, S. N. Chernikov, and N. V. Chernikova (Baeva). For complete factorizability, it is sufficient (Theorem 1) that each proper subgroup have a normal complement in some larger subgroup. A group is said to be weakly factorizable if each of its proper subgroups is complemented in some larger subgroup; the problem of describing finite groups with this property is posed (Question 8.31) in the “Kourovka Notebook”. Some properties of these groups are considered. The question is studied for Sylow $p$-subgroups of Chevalley-type groups of characteristic $p$. The main theorem, Theorem 2, establishes the weak factorizability of the Sylow $p$-subgroups in the symmetric and alternative groups and in the classical linear groups over fields of characteristic $p>0$, excluding the unitary groups of odd dimension $>p$.