Infinite groups with Abelian centralizers of involutions

Groups saturated by finite simple groups

“…The union of an ascending series of finite simple groups of Lie type of ranks bounded above is a simple group of Lie type over a locally finite field. PROPOSITION 1.12 [9]. Let G be a group containing a finite involution.…”

Periodic groups saturated with L 3(2m)

“…This 2-group is elementary Abelian since the Sylow 2-subgroup U/A of P is elementary Abelian. Now, by [9], P /C L 2 (Q) for some locally finite field Q of characteristic 2; in particular, P /C is locally finite. Groups C and A are locally finite, and so therefore is P .…”

Periodic groups saturated by finite simple groups U 3(2m)

“…The proposition is proved. [13]. Let G be a periodic group containing an involution and let the centralizer of every involution in G be elementary Abelian.…”