Locally finite groups with Černikov centralizers

On locally soluble periodic groups with Chernikov centralizer of a four-subgroup

“…If B t = l then B c Z d & v J ) . By Asar's result [1] Q/lQ,vJ is soluble so B centralizes some term of the derived series of Q. This shows that Q possesses a non-trivial characteristic abelian subgroup and we obtain a contradiction.…”

The Solution of a Problem of Kegel and Wehrfritz

“…The results of [1], [2], and [8] reduce the problem to the case of finite Sylow 2subgroups. So now a counter-example to the theorem can have only finite Sylow 2subgroups.…”

“…In [2] it was shown by the author that if the group G in the theorem is locally soluble then it is soluble. Subsequently the exact structure of the locally soluble G was determined by Hartley in [8].…”

“…Any element of R is a scalar multiple of one of the following types: |~0 s k l I" 0 s ' l = [i o) e3 = L-i 0} 0 Also w centralizes PGL(2, F o ). So it is sufficient to look at the centralizers of e x , e 2 , and e 3 in PGL(2,F 0 ). Let Z o = F* I.…”