Pálfy, Péter Pál
Published in
Periodica Mathematica Hungarica

For any finite solvable group G we show that if three primes dividing the degrees of certain irreducible characters of G are given, then there exists an irreducible character of G with degree divisible by at least two of the given primes.

Liu, Weijun Li, Huiling
Published in
Science in China Series A: Mathematics

Let S be a finite linear space, and letG be a group of automorphisms of S. IfG is soluble and line-transitive, then for a givenk but a finite number of pairs of (S, G),S hasv= pn points andG ⩽AΓL(1,pn).

Deryabina, G. S. Krasil'nikov, A. N.
Published in
Siberian Mathematical Journal

Given an arbitrary identity v=1, there exists a positive integer N=N(v) such that for every metabelian group G and every generating set A for G the following holds: If each subgroup of G generated by at most N elements of A satisfies the identity v=1 then the group G itself satisfies this identity. A similar assertion fails for center-by-metabelian...

Pittet, Ch. Saloff-Coste, L.
Published in
Journal of the European Mathematical Society

We establish the lower bound p2t(e,e)≿exp(-t1/3), for the large times asymptotic behaviours of the probabilities p2t(e,e) of return to the origin at even times 2t, for random walks associated with finite symmetric generating sets of solvable groups of finite Prüfer rank. (A group has finite Prüfer rank if there is an integer r, such that any of its...

Li, Xianhua Li, Shiheng
Published in
Siberian Mathematical Journal

We introduce the notion of θ-pair of a proper subgroup of a finite group and give necessary and sufficient conditions for supersolvability and nilpotence of a finite group. As application, we give a proof to a recent open problem of Skiba in the unsolved problems in group theory.

Gupta, Ch. K. Timoshenko, E. I.
Published in
Siberian Mathematical Journal

We study the first-order definable, Diophantine, and algebraic subsets in the set of all ordered sets generating a group or generating a group as a normal subgroup for some relatively free solvable groups.

Monakhov, V. S.
Published in
Mathematical Notes

Tests for π-solvability of a finite group with seminormal Hall π-subgroup are established and the nilpotency of the third commutator subgroup of any group with seminormal noncyclic Sylow subgroups is proved.

Knyagina, V. N. Monakhov, V. S.
Published in
Algebra and Logic

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB1 is a proper subgroup of G, for every proper subgroup B1 of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In part...

Guo, W. Shum, K. P. Skiba, A. N.
Published in
Siberian Mathematical Journal

Considering two subgroups A and B of a group G and ⊘ ≠ X ⊆ G, we say that A is X-permutable with B if ABx = BxA for some element x ∈ X. We use this concept to give new characterizations of the classes of solvable, supersolvable, and nilpotent finite groups.

Monakhov, V. S. Tyutyanov, V. N.
Published in
Siberian Mathematical Journal

We establish the solvability of each finite group whose every proper nonmaximal subgroup lies in some subgroup of prime index.