On a Free Action of a Group on an Abelian Group

Let x be an element of order 3 in a group G acting freely on a nontrivial abelian group. If for every gG the order of the commutator [x,g] is finite then x belongs to a finite normal subgroup of G.     

Minimal permutation representation of Thompson's simple group

Translated from Algebra i Logika, Vol. 27, No. 5, pp. 562–580, September–October, 1988.     

Finite groups with special Sylow 2-subgroups

Let T be a Sylow 2-subgroup of a simple group PSU (3, 2ⁿ), and Z a proper subgroup belonging to the center of T. We shall prove that a simple finite group whose Sylow 2-subgroup is isomorphic to T/Z coincides with PSU (3, 2ⁿ). As a consequence we list simple groups that can be represented in the form of a product of two Schmidt groups, i.e., of minimal nonnilpotent groups.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 217–222, August, 1973.     

A characterization of alternating groups

It is proved that a group G generated by a conjugacy class X of elements of order 3, so that every two non-commuting elements of X generate a subgroup isomorphic to an alternating group of degree 4 or 5, is locally finite. More precisely, either G contains a normal elementary 2-subgroup of index 3, or G is isomorphic to an alternating group of permutations on some (possibly infinite) set.Supported by RFBR grant Nos. 02-01-00495 and 02-01-39005, by FP Universities of Russia grant No. UR.04.01.0202, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 54–69, January–February, 2005.     

Finite Sylow Subgroups in Simple Locally Finite Groups of Lie Type

The main result of the paper is the followingTheorem. Let S = {r₀, r₁, ..., r_n} be a finite nonempty set of primes and let L be a Lie type of Chevalley groups. Then there exists a locally finite field F of characteristic r₀ such that Sylow r-subgroups of the simple group L(F) of type L over F are finite if and only if r ? S.     

2-Methyl-4-(o-nitrobenzylidene)oxazol-5-ones

The cis and trans isomers of 2-methyl-4-(o-nitrobenzylidene)oxazol-5-one have been obtained. On reduction, the cis isomer formed 3-acetamidoquinolin-2-ol, and the trans isomer formed 4-(o-aminobenzylidene)-2-methyloxazol-5-one. Details of the IR, UV, PMR, and mass spectra of the compounds obtained are given.Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 5, pp. 615–617 (1985).     

Finite Groups Close to Frobenius Groups

Siberian Mathematical Journal - We study finite nonsoluble generalized Frobenius groups; i.e., the groups G with a proper nontrivial normal subgroup F such that each coset Fx of prime order p, as...