Mahmut Kuzucuoğlu

Locally finite minimal non-FC-groups

We recall that a group G is an FC -group if for every x∈G the set of conjugates { x g |g∈G } is a finite set. Our interest here is with those groups G which are not FC groups while every proper subgroup of G is an FC -group: such groups are called minimal non-FC-groups. Locally finite minimal non- FC -groups with ( G ≠ G′ are studied in [1] and the structure of these groups is reasonably well understood. In [2] Belyaev has shown that a perfect, locally finite, minimal non- FC -group is either a simple group or a p -group for some prime p . Here we make use of the results of [5] to refine the result of Belyaev and provide a positive answer to problem 5·1 of [11]; in particular, we prove the following Theorem. There exists no simple, locally finite, minimal non-FC-group .

Locally Finite Barely Transitive Groups

Infinite transitive permutation groups all proper subgroups of which have just finite orbits are treated. Under the extra condition of being locally finite, such groups are proved to be primary, and, moreover, soluble if the stabilizer of some point is soluble.

Centralizers of Subgroups in Direct Limits of Symmetric Groups with Strictly Diagonal Embedding

Let ξ = (p 1, p 2,…) be a given infinite sequence of not necessarily distinct primes. In 1976, the structure of locally finite groups S(ξ) (respectively A(ξ) ) which are obtained as a direct limit of finite symmetric (finite alternating) groups are investigated in [7]. The countable locally finite groups A(ξ) gives an important class in the theory of infinite simple locally finite groups. The classification of these groups using the lattice of Steinitz numbers is completed by Kroshko and Sushchansky in 1998 see [8]. Here we extend the results on the structure of centralizers of elements to centralizers of arbitrary finite subgroups and correct some of the errors in the section of centralizers of elements in [8]. We construct for each infinite cardinal κ, a new class of uncountably many simple locally finite groups of cardinality κ as a direct limit of finitary symmetric groups. We investigate the centralizers of elements and finite subgroups in this new class of simple locally finite groups, and finally, we characterize this class by the lattice isomorphism with the cardinality of the group and the Steinitz numbers.

Description of Barely Transitive Groups with Soluble Point Stabilizer

We describe the barely transitive groups with abelian-by-finite, nilpotent-by-finite and soluble-by-finite point stabilizer. In article [6] Hartley asked whether there is a torsionfree barely transitive group. One consequence of our results is that there is no torsionfree barely transitive group whose point stabilizer is nilpotent. Moreover, we show that if the stabilizer of a point is a permutable subgroup of an infinitely generated barely transitive group G, then G is locally finite.

On locally graded barely transitive groups

We show that a barely transitive group is totally imprimitive if and only if it is locally graded. Moreover, we obtain the description of a barely transitive group G for the case G has a cyclic subgroup 〈x〉 which intersects non-trivially with all subgroups and for the case a point stabilizer H of G has a subgroup H1 of finite index in H satisfying the identity χ(H1) = 1, where χ is a multi-linear commutator of weight w.

CENTRALIZERS OF SUBGROUPS IN SIMPLE LOCALLY FINITE GROUPS

Abstract Hartley asked the following question: Is the centralizer of every finite subgroup in a simple non-linear locally finite group infinite? We answer a stronger version of this question for finite 𝒦-semisimple subgroups. Namely let G be a non-linear simple locally finite group which has a Kegel sequence 𝒦 = {(Gi, 1) : i ∈ ℕ} consisting of finite simple subgroups. Then for any finite subgroup F consisting of 𝒦-semisimple elements in G, the centralizer CG(F) has an infinite abelian subgroup A isomorphic to a direct product of ℤpi for infinitely many distinct primes pi. Moreover we prove that if G is a non-linear simple locally finite group which has a Kegel sequence 𝒦 = {(Gi, 1) : i ∈ ℕ} consisting of finite simple subgroups Gi and F is a finite 𝒦-semisimple subgroup of G, then CG(F) involves an infinite simple non-linear locally finite group provided that the finite fields ki over which the simple group Gi is defined are splitting fields for Li, the inverse image of F in Ĝi for all i ∈ ℕ. The group Ĝi is the inverse image of Gi in the corresponding universal central extension group.

Homogeneous monomial groups and centralizers

ABSTRACT The construction of homogeneous monomial groups are given and their basic properties are studied. The structure of a centralizer of an element is completely described and the problem of conjugacy of two elements is resolved. Moreover, the classification of homogeneous monomial groups are determined by using the lattice of Steinitz numbers, namely, we prove the following: Let λ and μ be two Steinitz numbers. The homogeneous monomial groups Σλ(H) and Σμ(G) are isomorphic if and only if λ = μ and H≅G provided that the splittings of Σλ(H) and Σμ(G) are regular.

Centralizers of Involutions in Locally Finite Groups

The present article deals with locally finite groups G having an involution φ such that C G (φ) is an SF-group. It is shown that G possesses a normal subgroup B which is a central product of finitely many groups isomorphic to PSL(2, K i ) or SL(2, K i ) for some infinite locally finite fields K i of odd characteristic, such that [G, φ]′/B and G/[G, φ] are both SF-groups.