A. S. Kondrat’ev

On finite tetraprimary groups

Chief factors of commutator subgroups of finite groups whose prime graph is disconnected and has exactly four vertices are described. As a corollary, finite simple groups recognizable by prime graph with exactly four vertices are determined.

On recognizability of some finite simple orthogonal groups by spectrum

It is proved that, if G is a finite group that has the same set of element orders as the simple group Dp(q), where p is prime, p ≥ 5 and q ∈ {2, 3, 5}, then the commutator group of G/F(G) is isomorphic to Dp(q), the subgroup F(G) is equal to 1 for q = 5 and to Oq(G) for q ∈ {2, 3}, F(G) ≤ G′, and |G/G′| ≤ 2.

A pronormality criterion for supplements to abelian normal subgroups

A subgroup H of a group G is called pronormal if, for any element g ∈ G, the subgroups H and Hg are conjugate in the subgroup . We prove that, if a group G has a normal abelian subgroup V and a subgroup H such that G = HV, then H is pronormal in G if and only if U = NU(H)[H,U] for any H-invariant subgroup U of V. Using this fact, we prove that the simple symplectic group PSp6n(q) with q ≡ ±3 (mod 8) contains a nonpronormal subgroup of odd index. Hence, we disprove the conjecture on the pronormality of subgroups of odd indices in finite simple groups, which was formulated in 2012 by E.P. Vdovin and D.O. Revin and verified by the authors in 2015 for many families of simple finite groups.

On recognizability by spectrum of finite simple groups of types Bn, Cn, and 2Dn for n = 2k

The spectrum of a finite group is the set of its element orders. A group is said to be recognizable (by spectrum) if it is isomorphic to any finite group that has the same spectrum. A nonabelian simple group is called quasi-recognizable if every finite group with the same spectrum possesses a unique nonabelian composition factor and this factor is isomorphic to the simple group in question. We consider the problem of recognizability and quasi-recognizability for finite simple groups of types Bn, Cn, and 2Dn with n = 2k.

On the behavior of elements of prime order from Singer cycles in representations of special linear groups

Let G = SLn(q), where n ≥ 2 and q is a power of a prime p. A Singer cycle of the group G is its cyclic subgroup of order (qn − 1)/(q − 1). We classify absolutely irreducible G-modules over a field of characteristic p where an element of fixed prime order m from a Singer cycle of G acts freely in the following three cases: (a) the residue of q modulo m generates the multiplicative group of the field of order m (in particular, this holds for m = 3); (b) m = 5; (c) n = 2.

The Complete Reducibility of Some GF (2)A 7 -Modules

It is proved that, if G is a finite group with a nontrivial normal 2-subgroup Q such that G/Q ∼= A7 and an element of order 5 from G acts freely on Q, then the extension G over Q is splittable, Q is an elementary abelian group, and Q is the direct product of minimal normal subgroups of G each of which is isomorphic, as a G/Q-module, to one of the two 4-dimensional irreducible GF(2)A7-modules that are conjugate with respect to an outer automorphism of the group A7.

Finite Groups Having the Same Prime Graph As the Group Aut(J2)

Finite groups having the same prime graph as the group Aut(J2) are described. This solves the problem posed by B. Khosravi.

Synthesis of β-functionalized ethyl polyfluoroaryl sulfides, sulfoxides, and sulfones underlain by pentafluorobenzoic acid

The reaction of pentafluorobenzoic acid with 2-mercaptoethanol followed by modification of the reaction product by oxidationm halohydroxylation and (or) decarboxylation led to the formation of β-halo- and β-oxyethyl polyfluoroaryl sulfides, sulfoxides, and sulfones, initial compounds for the synthesis of fluorinecontaining 2,3-dihydro-1,4-benzothiazines, 2,3-dihydro-1,4-benzoxathiynes, and 2,3-dihydro-1,4-benzodithiynes and also of their S-oxides and S-dioxides based on intramolecular nucleophilic sybstitution.

Stabilizers of vertices of graphs with primitive automorphism groups and a strong version of the Sims conjecture. I

This is the first in a series of papers whose results imply the validity of a strengthened version of the Sims conjecture on finite primitive permutation groups from the authors’ article “Stabilizers of graph’s vertices and a strengthened version of the Sims conjecture”, Dokl. Math. 59 (1), 113–115 (1999). In this paper, the case of not almost simple primitive groups and the case of primitive groups with alternating socle are considered.

Finite groups whose prime graphs do not contain triangles. I

Finite groups whose prime graphs do not contain triangles are investigated. In the present part of the study, the isomorphic types of prime graphs and estimates of the Fitting length of solvable groups are found and almost simple groups are determined.