Locally finite groups containing a $2$-element with Chernikov   centralizer
  Suppose that a locally finite group $G$ has a $2$-element $g$ with Chernikov centralizer. It is proved that if the involution in $\langle g\rangle$ has nilpotent centralizer, then $G$ has a soluble subgroup of finite index. 
E. I. Khukhro, N. Yu. Makarenko, and P. Shumyatsky
math.GR