On locally finite minimal non-solvable groups

In the present work we consider infinite locally finite minimal non-solvable groups, and give certain characterizations. We also define generalizations of the centralizer to establish a result relevant to infinite locally finite minimal non-solvable groups.     

On locally primitive Cayley graphs of finite simple groups

In this paper we investigate locally primitive Cayley graphs of finite nonabelian simple groups. First, we prove that, for any valency d for which the Weiss conjecture holds (for example, d?20 or d is a prime number by Conder, Li and Praeger (2000) [1]), there exists a finite list of groups such that if G is a finite nonabelian simple group not in this list, then every locally primitive Cayley graph of valency d on G is normal. Next we construct an infinite family of p-valent non-normal locally primitive Cayley graph of the alternating group for all prime p?5. Finally, we consider locally primitive Cayley graphs of finite simple groups with valency 5 and determine all possible candidates of finite nonabelian simple groups G such that the Cayley graph Cay(G,S) might be non-normal.     

Examples of locally finite groups

In this paper we construct examples of nonextremal locally finite groups for which every finite unsolvable subgroup is not embedded in any proper infinite subgroup. We also give an example of a locally finite simple group, any proper infinite subgroup of which is bigrade solvable.Translated from Matematicheskie Zametki, Vol. 13, No. 1, pp. 103–106, January, 1973.The author wishes to express his deep gratitude to Yu. M. Gorchakov for help and valuable advice in the preparation of the paper.     

Positive universal classes in locally finite varieties

In some recent papers, the concept of a Q-independent sequence of finite lattices was utilized. We investigate this concept in universal algebras and apply it to positive universal classes in locally finite varieties, with emphasis on semilattices, lattices, and their expansions.     

On some locally finite groups which are sharply triply transitive

We classify the groups satisfying the following conditions: i) is locally finite; ii) is a sharply triply transitive permutation group; iii) all elements of have fixpoints.Published in Ukraninskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp. 1060–1065, July–August, 1991.     

Dual-standard subgroups of finite and locally finite groups

A subgroup D of a group G is called dual-standard if, for all subgroups X and Y of G, <XχD,YχD>=<X,Y>χD. When G is finite, Zappa has given some information concerning the way in which D is embedded in G and the structure of G itself. Among other things, Zappa makes reference to the maximal normal Hall subgroup L of G contained in D. In general L can be arbitrary. The main result of this work, however, is to show that the commutator of L with a complement in G is nilpotent.     

Topologization of countable locally finite groups

It is shown that every countable locally finite group admits the antidiscrete Hausdorff group topology.Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 613-618, November-December, 1995.Supported by the Krasnoyarsk Territory Science Foundation, grant No. 3F0156.