On a Sufficient Condition when an Infinite Group Is not Simple

V. P. Shunkov in [14] proved his famous theorem on the local finiteness and almost solvability of a periodic group G containing an involution with a finite centralizer. V. V.Belyaev in [2] on the basis of ideas from the work of V.P. Shunkov proved that any group G containing a finite involution z with a finite centralizer is locally finite. The finiteness of the involution z means that the group ⟨z, z⟩ is finite for any g ∈ G. A. I. Sozutov in [11] showed, in particular, that any group G, containing an almost perfect involution z with a finite centralizer, is not simple. An involution z of a group G is said to be almost perfect if from the condition |zz| = ∞, where g ∈ G, implies the equality z = z for some involution x from G. In all the above papers, [2, 11, 14] it was shown that the group G is not simple (under the assumption that G is an infinite group). It was natural to consider the situation when the group G contains an involution z such that CG(z) contains a finite number of elements of finite order, but CG(z) does not have to be finite, unlike the groups from the papers [2, 11,14].