X-quasinormal subgroups

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

Fitting Classes with Given Properties of Hall Subgroups

In the theory of formations of finite solvable groups, there is a well-known result due to Blessenohl claiming that, for any local formation , the class of groups for which every Hall π-subgroup belongs to also is a local formation. In the present paper, we obtain a result exactly dual to that indicated in the theory of Fitting classes. We prove that if a Fitting class is local, then the class of all groups all of whose Hall π-subgroups belong to is also local.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 234–240.Original Russian Text Copyright © 2005 by V. N. Zagurskii, N. T. Vorob’ev.     

有限交换群的自同构群阶

初等交换P-群的自同构群阶已经得到,对于其它情形则鲜有结果.文中得到了2类有限交换群的自同构群阶,并推广了P.Hall的一个相关结果.       

pnqm(m≤2)阶群的可解性与结构

利用Sylow定理证明P^nq^m(m≤2)阶群是可解群而且给出了它的结构．     

A new characterization of L 2(r) by their Sylow numbers

Let G be a finite centerless group, let π(G) be the set of prime divisors of the order of G, and let np(G) be the number of Sylow p-subgroups of G, that is, n_p(G) = |Sylp(G)|. Set NS(G) := {n_p(G)| p ∈π(G)}. In this paper, we are investigating whether L_2(r) is determined up to isomorphism by NS(L_2(r)) when r is prime.     

Unfused Involutions in Finite Groups: An Addendum

If p is a prime, r is an integer and p ^r  > 2, there exist infinitely many finite simple non-Abelian groups containing an element x of order p ^r such that if P is a Sylow p-subgroup of G containing x, x ^G  ∩ P = x ^P .     

On c-Normal Subgroups of Finite Groups

Let G be a finite group. We fix in every noncyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P| and study the structure of G under the assumption that all subgroups H of P with |H| = |D| are c-normal in G.     

On a Finite Group Having a Normal Series Whose Factors Have Bicyclic Sylow Subgroups

We consider the structure of a finite group having a normal series whose factors have bicyclic Sylow subgroups. In particular, we investigate groups of odd order and A ₄-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.