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.

No abstract

Denote by M the set whose elements are the simple 3-dimensional unitary groups U 3 (q) and the linear groups L 3 (q) over finite fields. We prove that every periodic group, saturated by the groups of a finite subset of M, is finite.Keywords: saturation of a group by a set of groups, periodic group Introduction. A group G is called saturated by groups in some set R if each finite subgroup of G is contained in a subgroup of G isomorphic to some group in R. Take some group G saturated by groups in R and suppose that to X ∈ R there is a subgroup L of G isomorphic to X. In this case we say that G is saturated by R and call R a saturating group set for G.It was conjectured [1] that each periodic group is finite that is saturated by finitely many nonabelian simple groups. This conjecture was confirmed in the case that every element of a saturating group set R is a finite simple group the centralizer of whose Sylow 2-subgroup contains no elements of odd order greater than 3 [1], and also in the casesDenote by δ the variable taking the values + and −. Denote by L δ 3 (p n ) the group L 3 (p n ) if δ = + and the group U 3 (p n ) if δ = −.

Computational group theory investigates the arithmetic properties of algebraic structures. These properties may be utilized in development of algorithms for data analysis and machine learning. This direction is associated with the design, analysis of algorithms and data structures for calculating various characteristics for (mostly finite) groups. The area is interesting for investigation of important from various points of view groups that are important, the data about which cannot be obtained by manually. The present work was carried out in line with computer calculations in groups. Based on the concept of the group growth function, the concept of the group density function is introduced. The growth and density functions of finite simple non-Abelian groups of small orders are constructed. The question of recognizability of finite simple non-Abelian groups by the density function is considered. The problem of effective storage of finite group elements in computer memory using the AVL tree is investigated. Based on the AVL tree, a fast algorithm for enumerating all elements of a finite group is developed. With the computer calculations, the recognizability of two finite simple non-Abelian groups by the group density function is proved.

We give a full description of periodic groups with given Sylow 2-subgroups saturated by a set of finite simple groups.