We prove the existence of infinite subgroups with nontrivial locally finite radicals and of locally finite subgroups in the groups with almost finite almost solvable elements of prime orders and in the groups with generally finite elements.In this article we generalize some results of [1-10] and prove the theorems that are announced in [11,12].If the set of finite order elements is finite in an infinite group G then by the well-known theorem of Dietzmann this set forms a finite fully characteristic subgroup of G. If the set of finite order elements in G is infinite then some questions concerning their location in the group are in order [9]. One of them is the well-known question of Kargapolov [13, Problem 1.24] about the existence of infinite abelian subgroups in an infinite group, which in the general case was answered negatively by Novikov and Adyan [14]. Therefore, it has to be considered separately for many classes of infinite groups. The question of Kargapolov is closely related to a weaker question of the existence of the f -local subgroups, i.e., the infinite subgroups with nontrivial locally finite radicals (see the question of Strunkov [13, Problem 2.75]). In this article we give positive answers to these questions in some classes of groups.Theorem 1. Given an infinite group G, take an element a of prime order p > 2 so that for almost all a g ∈ a G the subgroups a, a g are finite and solvable. Then either G contains only finitely many finite order elements or a lies in an f -local subgroup that contains infinitely many finite order elements.Call an element x of a group G solvable whenever all finite subgroups generated by elements in x G are solvable.Theorem 2. In an infinite group G take solvable elements a and b of prime orders so that |a|·|b| > 4 and for almost all c ∈ b G the subgroups a, c are finite. Then either G contains only finitely many finite order elements or at least one of a and b lies in an f -local subgroup that contains infinitely many finite order elements.A group G is said to possess periodic part whenever the set of finite order elements of G forms a subgroup [9]. Call a finite order nonidentity element a of some infinite group G generally finite if a lies in the base of a fan of finite subgroups whose amalgam almost completely contains a nonidentity conjugacy class of G. In other words, for almost all c in some nonidentity conjugacy class b G the subgroups a, c are finite; i.e., the (a, b)-finiteness condition [9] holds.Theorem 3. If all finite subgroups of G are solvable and each quotient of G by a finite subgroup either is torsion free or contains a generally finite element of odd prime order then G either possesses finite periodic part or includes an infinite locally finite subgroup.In the case that the general finiteness holds for all prime order elements in G and is inherited by all subgroups of G and all its quotients by periodic normal subgroups, call G a generally finite group. Theorem 3 implies Corollary 1. If the set of finite order elements in a generally finite group G i...

We prove the existence of infinite subgroups with nontrivial locally finite radicals and of infinite locally finite subgroups in the groups with almost finite almost solvable elements of order 2 and 4 and in the groups with almost finite elements. This article is a continuation of [1]. Recall that by an f -local subgroup of an infinite group G we mean every infinite subgroup with a nontrivial locally finite radical [2]. We say that a mixed group G possesses a periodic part if all finite order elements of G constitute a subgroup [3]. A nontrivial finite order element a of some infinite group G is called generalized finite if for almost all elements b g of some conjugacy class b G , where b = 1, the subgroups a, b g are finite; i.e., the (a, b)-finiteness condition [3] is fulfilled. If in addition b ∈ a G then the element a is called almost finite. The results of [1] and this article yield sufficient conditions for the existence of f -local and infinite locally finite subgroups in the groups with almost finite and generalized finite elements (Questions 1.24 by Kargapolov and 2.75 by Strunkov [4]). If the (a, b)-finiteness condition is fulfilled in some group then it holds for some pair of prime order elements. As Shunkov showed [5], the case |a| = |b| = 2 and b ∈ a G is special: in this case any analog of Theorem 1 fails; see Example 1 and Theorem 1 in [1]. However, if b / ∈ a G then both involutions a and b belong to f -local subgroups that contain infinitely many finite order elements [6]. Therefore, the cases of an order 4 almost finite element and a pair (a, b) with |a| · |b| = 8 emerge as subjects of independent investigation, which is presented in this article.Theorem 1. Take some infinite group G, some order 4 element a in G, and suppose that for almost all elements a g ∈ a G the subgroups a, a g are finite and solvable. Then either G possesses finite periodic part or a belongs to some f -local subgroup that contains infinitely many finite order elements.Theorem 2. Take some infinite group G and some elements a, b in G, one of which is an involution and the other has order 4, and suppose that for almost all elements b g ∈ b G the subgroups a, b g are finite and solvable. Then either G possesses finite periodic part or at least one of a and b belongs to some f -local subgroup that contains infinitely many finite order elements.Theorem 3. Given some group G, if all finite subgroups are solvable and every quotient by a finite subgroup either is a torsion-free group or contains a generalized finite element of order > 2 then G either possesses finite periodic part or includes an infinite locally finite subgroup.

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