Given an indexing set I and a finite field K α for each α ∈ I, let R = {L 2 (K α ) | α ∈ I} and N = {SL 2 (K α ) | α ∈ I}. We prove that each periodic group G saturated with groups in R (N) is isomorphic to L 2 (P ) (respectively SL 2 (P )) for a suitable locally finite field P .A group G is saturated with groups in a set M [1] whenever each finite subgroup K of G lies in some subgroup L of G isomorphic to a group in M. If G is saturated with groups in M, and for each X ∈ M there is a subgroup L of G isomorphic to X, then G is saturated with M, and M itself is called a saturating set of groups for G.In this article I denotes an indexing set. Given a finite field K α for eachNote that for distinct α and β the characteristics of K α and K β may be different.We prove the following results.Theorem 1. An infinite periodic group G saturated with groups in R is isomorphic to the simple group L 2 (P ) over a suitable locally finite field P .Theorem 2. An infinite periodic group G saturated with groups in N is isomorphic to the group SL 2 (P ) over a suitable locally finite field P .Proof of Theorem 1. Suppose that G is a counterexample to Theorem 1. Denote by S a Sylow 2-subgroup of G. Lemma 1. All involutions in G are conjugate.Proof. Take two distinct involutions x and y in G. The saturation condition gives x, y ⊂ L ⊂ G, where L L 2 (K α ). It is well known [2] that in L 2 (K α ) all involutions are conjugate. Thus, x = y g for some g ∈ L, which proves the lemma.Lemma 2. If S is a finite group then all Sylow 2-subgroups of G are conjugate, and S is of one of the following types:(1) the dihedral group;(2) an elementary abelian group with |S| > 4.Proof. The fact that all Sylow 2-subgroups of G are finite and conjugate follows from [3]. The saturation condition gives S ⊂ L ⊂ G and L L 2 (K α ). By [2, pp. 9-10] S is either of type 1 or of type 2. The lemma is proved.Lemma 3. If S is an infinite group then all Sylow 2-subgroups of G are conjugate, and S is of one of the following types:(where S is a quasicyclic 2-group, t 2 = 1 and s t = s −1 for each s ∈ S; (4) an infinite elementary abelian group. Proof. Suppose first that G contains no elements of order 4. Then each 2-subgroup of G, and in particular S, is elementary abelian. Take another Sylow 2-subgroup S 1 in G. Take involutions x ∈ S and y ∈ S 1 . The saturation condition gives x, y ⊂ R L 2 (K α ). Consequently, x = y g for some g ∈ R andKrasnoyarsk.

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We deal with periodic groups saturated with dihedral groups. In particular, it is proved that periodic groups of bounded period, and also periodic Shunkov groups, saturated with dihedral groups, are locally finite. DEFINITIONS AND MAIN RESULTSWe say that a group G is saturated with groups from a set R if every finite subgroup K of G is contained in a subgroup isomorphic to some group of R (see [1]). If G is saturated with groups from a set R, and for any X ∈ , G contains a subgroup L X, then we say that G is saturated with the set R of groups, and we call R a saturating set of groups for G.Recall that a group generated by two involutions is called a dihedral group, or dihedron. And if such a group is finite, then we refer to it as a finite dihedron. A locally finite dihedron is a group that is a union of an infinite ascending chain of finite dihedrons. A Shunkov group is one in which every pair of conjugate elements of prime order generate a finite group, with this property preserved over finite sections.In the present paper we prove the following theorems. THEOREM 1.A periodic Shunkov group saturated with dihedral groups is locally finite. THEOREM 2.A periodic group of bounded period saturated with dihedral groups is finite. THEOREM 3.If G is a periodic group saturated with dihedral groups and S is its Sylow 2-subgroup, then either S is a group of order 2 and G is a (locally) finite dihedron, or G = ABC = ACB = BCA = CBA where A is the centralizer of some involution z in the center of S, B = O(C G (v)) where v is an arbitrary involution in S, distinct from z, and C = O(C G (zv)). Moreover, A is a (locally) finite dihedron and B, C are (locally) cyclic groups. PRELIMINARIESLEMMA 1. A locally finite dihedron G is generated by involutions. Proof. Indeed, G is generated by finite dihedral groups, which are generated by involutions; so, G is generated by involutions. The lemma is proved.LEMMA 2. Let G be a locally finite dihedron. Then G = Hλ i , where H is a locally cyclic group, i is an involution, and for any element h ∈ H, h i = h −1 . In particular, every (locally) cyclic subgroup of G, *