Solvable Supplements to Normalizers of Cyclic 2-Subgroups

Amberg and Kazarin proved that a finite group is solvable if the normalizer of every cyclic subgroup of prime power order has a solvable supplement. We substantially relax this hypothesis by requiring it only for cyclic $2$-subgroups. This condition, denoted by ${\mathrm{S}\mathrm{S}\mathrm{N}}_2$, sharply restricts the nonabelian composition factors of the group to the family ${\text{PSL}}_2(q)$, where $q\ge 7$ is a prime power satisfying $q\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$. Conversely, this family is precisely the nonabelian finite simple groups that satisfy ${\mathrm{S}\mathrm{S}\mathrm{N}}_2$. Consequently, a finite group satisfying ${\mathrm{S}\mathrm{S}\mathrm{N}}_2$ is solvable if and only if it has no section isomorphic to one of these groups.