p-Frattini子群与P-中心

本文对p-Frattini子群进行了进一步的研究,给出了p-Frattini子群与p-中心的关系;我们的结果推广了Gaschutz在文献[1]中关于Frattini子群与群的中心间的关系.     

域上Cl型Chevalley群中含单项子群的极大子群

对特征不为2的任意域上Ci型Chevaley群,构造了一类真包含单项子群的子群,从而否定回答子单项子群的极大性问题,同时证明了所构造的群恰为极大子群。     

A New Type of Pronormal Subgroups in Chevalley Groups

Let L be a simple Lie algebra with irreducible root system φ having roots of different length,F be a field of characteristic different from 2,G=L(F) be a Chevalley group of type L over F.Denote by φ^1 the set of all long roots in φ.Set G^1=(zr(t);r∈φ^t,t∈F).It is a subgroup of G generated by all the long root subgroups.This paper determines the pronormality of G^1 in G when L is not of type G2.     

The signatures of the congruence subgroups G 0(τ) of the hecke groups G 4 and G 6

We determine the signatures of the congruence subgroups of the Hecke groups G ₄ and G ₆.     

Groups with π-maximal and π-layer maximal conditions

By analogy with π-minimal, π-layer minimal, primary minimal, and layer minimal conditions, similar maximal conditions are introduced and studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 311–320, August, 2000.     

On the Size of Boolean Combinations of Subgroups of Finite Abelian Groups

The sizes of Boolean combinations of subgroups G _i of a finite abelian group depends only on the Boolean expression, the 0-1-sublattice generated by the G _i , and the size of minimal subquotients from this sublattice. Moreover, they increase, monotonically, with those sizes.     

Finite Groups with Few TI-Subgroups

A subgroup A of a finite group G is called a TI-subgroup if either A ∩ A ^x  = 1 or A ∩ A ^x  = A holds for all x ∈ G. In this paper, finite group all of whose meta-cyclic subgroups are TI-subgroups are classified completely. In particular, such groups are solvable.     

DECIMAL EXTENSIONS OF THE ADDITIVE GROUP OF INTEGERS

We introduce the concept of decimal extension of the additive group of integers. We characterize these extensions as the linearly ordered abelian groups containing at least one Archimedean element. Finally, we study the additive group of rationals and its subgroups as particular cases of decimal extension.     

An algorithm and self-reproducing systems

Let G be a group and let N be a normal subgroup of G. We set cd(G|N) to be the degrees of the irreducible characters of G whose kernels do not contain N. We associate a graph with this set. The vertices of this graph are the primes dividing degrees in cd(G|N), and there is an edge between p and q if pq divides some degree in cd(G|N). In this paper, we study this graph when it is disconnected, and we study its diameter when it is connected.     

Groups with a Maximal Irredundant 6-Cover

ABSTRACT

A cover for a group G is a collection of proper subgroups whose union is the whole group G. A cover is irredundant if no proper sub-collection is also a cover, and is called maximal if all its members are maximal subgroups. For an integer n > 2, a cover with n members is called an n-cover. Also, we denote σ (G) = n if G has an n-cover and does not have any m-cover for each integer m < n. In this article, we completely characterize groups with a maximal irredundant 6-cover with core-free intersection. As an application of this result, we characterize the groups G with σ (G) = 6. The intersection of an irredundant n-cover is known to have index bounded by a function of n, though in general the precise bound is not known. We also prove that the exact bound is 36 when n is 6.