On Generation of Sporadic Simple Groups by Three Involutions Two of Which Commute

We prove the following result: Let G be one of the 26 sporadic simple groups. The group G cannot be generated by three involutions two of which commute if and only if G is isomorphic to M ₁₁, M ₂₂, M ₂₃, or M ^cL .     

Characterization of Locally Finite Simple Groups of Type G2 over Fields of Odd Characteristics in the Class of Periodic Groups

Mathematical Notes - We prove that a periodic group is locally finite, given that each of its finite subgroups lies in a subgroup isomorphic to a finite simple group G2 of Lietypeovera field of odd...     

Characterizations of finite groups by sets of orders of their elements

For a finite group G, ω(G) denotes the set of orders of its elements. If ω is a subset of the set of natural numbers, h(ω) stands for the number of pairwise nonisomorphic finite groups G for which ω(G)=ɛ. We prove that h(ω(G))=1, if G is isomorphic to S₉, S₁₁, S₁₂, S₁₃, or A₁₂, and h(ω(G))=2 if G is isomorphic to S₂(6) or to O ₈ ⁺ (2). 01 Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 37–53, January–February, 1997.     

Locally Finite Periodic Groups Saturated with Finite Simple Orthogonal Groups of Odd Dimension

Siberian Mathematical Journal - Suppose that $ n $ is an odd integer, $ n\geq 5 $ . We prove that a periodic group  $ G $ , saturated with finite simple orthogonal groups  $...     

Periodic groups saturated with <Emphasis Type="Italic">L</Emphasis><Subscript>3</Subscript>(2<Superscript>m</Superscript>)

Let be a set of finite groups. A group G is saturated with groups from if every finite subgroup of G is contained in a subgroup isomorphic to some member of . It is proved that a periodic group G saturated with groups from the set {L₃(2^m)|m = 1, 2, …} is isomorphic to L₃(Q), for a locally finite field Q of characteristic 2; in particular, it is locally finite. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 606–626, September–October, 2007.     

Element orders in coverings of symmetric and alternating groups

We prove that if the factor group H=G/N of a finite group G is isomorphic to a symmetric or alternating group of degree m, where m≥5 and N≠1, then G has an element whose order is distinct from any element’s order in H. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 296–315, May–June, 1999.     

Recognition of finite simple groupsL 3(2 m ) ANDU 3(2 m ) by their element orders

It is proved that a finite group isomorphic to a simple non-Abelian group L₃(2^m) or U₃(2^m) is, up to isomorphism, recognizable by a set of its element orders. On the other hand, for every simple group S=S₄(2^m), there exist infinitely many pairwise non-isomorphic groups G with w(G)=w(S). As a consequence, we present a list of all recognizable finite simple groups G, for which 4t ∉ ω(G) with t>1. Supported by RFFR grant No. 99-01-00550, by the National Natural Science Foundation of China (grant No. 19871066), and by the State Education Ministry of China (grant No. 98083). Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 567–585, September–October, 2000.