Periodic Groups Saturated with the Groups L 2(p n )

Abstract

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.

Periodic groups saturated with L 3(2m)

“…A group G is said to be saturated with groups from M if every finite subgroup of G is contained in a subgroup isomorphic to some member of M. In [1,Question 14.101], it was conjectured that every periodic group saturated with groups from the set of finite simple groups of Lie type, whose ranks are bounded in totality, is isomorphic to a simple group of Lie rank over a locally finite field. In [2][3][4], this conjecture was confirmed for the cases where M consists of, respectively, Ree groups, projective special linear groups of dimension 2, and Suzuki groups.…”

On the periodic groups saturated with projective linear groups

“…Also in [3] this result was extended to the case when a group is saturated with SL 2 (q). It is natural to consider the case when a periodic group is saturated with groups in M = {P GL 2 (q) | q is a prime power}.…”

“…Suppose that I is a nonempty set of indices, K α is a finite field for every α ∈ I, and R = {L 2 (K α ) | α ∈ I}. A periodic group G saturated with groups in R is isomorphic to the simple group L 2 (P ) over a suitable locally finite field P [3]. Proposition 4.…”

“…It was proved in [3] that each periodic group saturated with the groups L 2 (q), where q is not fixed, is isomorphic to L 2 (Q), where Q is a locally finite field. Also in [3] this result was extended to the case when a group is saturated with SL 2 (q).…”

Periodic groups with given properties of finite subgroups

“…It has been proved in [4] that a periodic group sat urated with finite simple groups L 2 (q), is isomorphic to a group L 2 (Q) for some locally finite field Q. We gener alize this result as follows.…”