Romanovskii, N. S. Timoshenko, E. I.
Published in
Siberian Mathematical Journal

Considering the partially commutative groups of the varieties that include the variety of all length 2 nilpotent groups, we study the questions of factor cancellation in direct products, coincidence of elementary theories, characterization of the group by its elementary theory, and the possibility of direct product decomposition.

Vasil’ev, A. F. Murashka, V. I.
Published in
Mathematical Notes

A subgroup H of a finite group G is said to be F(G)-subnormal if it is subnormal in HF(G), where F(G) is the Fitting subgroup of G. In the paper, the problem of whether or not a formation β contains products of F(G)-subnormal β-subgroups of finite solvable groups is studied. In particular, solvable saturated formations β with this property are desc...

Timoshenko, E. I.
Published in
Algebra and Logic

We study elementary and universal theories of relatively free solvable groups in a group signature expanded by one predicate distinguishing primitive or annihilating systems of elements.

Gorshkov, I. B. Maslova, N. V.
Published in
Algebra and Logic

It is shown that the Gruenberg–Kegel graph of a finite almost simple group is equal to the Gruenberg–Kegel graph of some finite solvable group iff it does not contain 3-cocliques. Furthermore, we obtain a description of finite almost simple groups whose Gruenberg–Kegel graphs contain no 3-cocliques.

Roman’kov, V. A.
Published in
Algebra and Logic

A verbal subset of a group G is a set w[G] of all values of a group word w in this group. We consider the question whether verbal subsets of solvable groups are rational in the sense of formal language theory. It is proved that every verbal subset w[N] of a finitely generated nilpotent group N with respect to a word w with positive exponent is rati...

Krasnikov, A. F.
Published in
Lobachevskii Journal of Mathematics

Let F be a free group with basis {xj|j ∈ J}; N a normal subgroup of F. For a given element n of N we describe an elements Dl(n), where Dl: Z(F) → Z(F) (l ∈ J) are the Fox derivations of the group ring Z(F). If r1, r2 are an elements of F/[N,N] and, for some positive integer d, r1d is in the normal closure of r2d in F/[N,N], then r1 is in the normal...

Baikalov, A. A.
Published in
Algebra and Logic

It is proved that for any solvable subgroup G of an almost simple group S with simple socle isomorphic to An, n ≥ 5, there are elements x, y, z, t ∈ S such that G ∩ Gx ∩ Gy ∩ Gz ∩ Gt = 1.

Alekseeva, O. A. Kondrat’ev, A. S.
Published in
Proceedings of the Steklov Institute of Mathematics

The study of finite groups whose prime graphs do not contain triangles is continued. The main result of this paper is the following theorem: if G is a finite nonsolvable group whose prime graph contains no triangles and S(G) is the greatest solvable normal subgroup of G, then |π(G)| ≤ 8 and |π(S(G))| ≤ 3. A detailed description of the structure of ...

Meng, Wei Yao, Hailou
Published in
Indian Journal of Pure and Applied Mathematics

Let G be a finite group and NA(G) denote the number of conjugacy classes of all nonabelian subgroups of non-prime-power order of G. The Symbol π(G) denote the set of the prime divisors of |G|. In this paper we establish lower bounds on NA(G). In fact, we show that if G is a finite solvable group, then NA(G) = 0 or NA(G) ≥ 2|π(G)|−2, and if G is non...

Alekseeva, O. A. Kondrat’ev, A. S.
Published in
Proceedings of the Steklov Institute of Mathematics

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.