Fusion of $2$-elements in periodic groups with finite Sylow $2$-subgroups

This article contributes to the study of a fusion of subsets in finite Sylow $2$-subgroups of periodic groups. We extend well-known theorems on fusion of subsets in Sylow subgroups of finite groups by Burnside and Alperin to periodic groups which contain a finite Sylow $2$-subgroup.