Sum of Element Orders of Maximal Subgroups of the Symmetric Group

For a finite group G, let ψ(G) denote the sum of element orders of G. The aim of this article is to show that ψ(H) < ψ(A _n ) for every proper subgroup H of the symmetric group of degree n, which is different from the alternating group A _n .     

Quasi-Purifiable Subgroups and Minimal Direct Summands

Let G be an arbitrary Abelian group. A subgroup A of G is said to be quasi-purifiable in G if there exists a pure subgroup H of G containing A such that A is almost-dense in H and H/A is torsion. Such a subgroup H is called a “quasi-pure hull” of A in G. We prove that if G is an Abelian group whose maximal torsion subgroup is torsion-complete, then all subgroups A are quasi-purifiable in G and all maximal quasi-pure hulls of A are isomorphic. Every subgroup A of a torsion-complete p-primary group G is contained in a minimal direct summand of G that is a minimal pure torsion-complete subgroup containing A. An Abelian group G is said to be an “ADE decomposable group” if there exist an ADE subgroup K of G and a subgroup T′ of T(G) such that G = K⊕ T′. An Abelian group whose maximal torsion subgroup is torsion-complete is ADE decomposable. Hence direct products of cyclic groups are ADE decomposable groups.     

A Note on Projective and Flat Dimensions and the Bieri-Neumann-Strebel-Renz Σ-Invariants

Let G be a finitely generated group, and A a ?[G]-module of flat dimension n such that the homological invariant Σ ⁿ (G, A) is not empty. We show that A has projective dimension n as a ?[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Σ ⁿ (G, ?) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd (G) = hd(G) implies that there is a subgroup H of finite index in G such that Σ^∞(H, ?) ≠ ?.     

Nearly m-embedded subgroups and solvability of finite groups

A subgroup A of a finite group G is called {1≤G}-embedded in G if for each two subgroups K≤H of G, where K is a maximal subgroup of H, A either covers the pair (K,H) or avoids it. Moreover, a subgroup H of G is called nearly m-embedded in G if G has a subgroup T and a {1≤G}-embedded subgroup C such that G?=?HT and H∩T≤C≤H. In this paper, we mainly prove that G is solvable if and only if its Sylow 3-subgroups, Sylow 5-subgroups and Sylow 7-subgroups are nearly m-embedded in G.     

On ℑ<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-normal subgroups of finite groups

Given a class ℑ of finite groups, a subgroup H of a group G is called ℑ _n -normal in G if there exists a normal subgroup T of G such that HT is a normal subgroup of G and (H ∩ T)H _G /H _G is contained in the ℑ-hypercenter Z _∞^ℑ (G/H _G ) of G/H _G . We obtain some results about the ℑ _n -normal subgroups and use them to study the structure of some groups.     

θ-Pairs for 2-maximal subgroups of finite groups

Let H, A and B be subgroups of a group G. We call the pair (A, B) a θ-pair for H in G if: (i) áH, A? = G{\langle H, A\rangle=G} and B = (A ∩ H) _G ; (ii) if A ₁/B is a proper subgroup of A/B and A₁/B \vartriangleleft G/B{{A_1/B \vartriangleleft G/B}}, then G 1 áH, A₁?{G\neq \langle H, A_1\rangle}. In this paper, we study the θ-pairs for 2-maximal subgroups of a group, which imply a group to be solvable or supersolvable.     

Some Notes on Meet-Irreducible Subgroups of Finite Groups

A proper subgroup A of a finite group G is said to be primitive or meet-irreducible if there is a unique subgroup A₀ ≤ G such that A is a maximal subgroup of A₀. In this case we say that |A₀: A| is the small index of A and denote it by |G: A|₀. In this article, we study the influence of meet-irreducible subgroups and their small indexes on the structure of G. In particular, we prove that a finite group G is supersoluble if and only if |G: A|₀ = |G: B|₀ for any two meet-irreducible subgroups A and B of G with A_G = B_G.     

On Modules with Few Minimax Cocentralizers

Let R be a ring and G a group. An R-module A is said to be minimax if A includes a noetherian submodule B such that A/B is artinian. The authors study a ?G-module A such that A/C _A (H) is minimax (as a ?-module) for every proper not finitely generated subgroup H.     

π-quasinormally embedded and c-supplemented subgroup of finite group

Suppose that H is a subgroup of a finite group G. H is called π-quasinormal in G if it permutes with every Sylow subgroup of G; H is called π-quasinormally embedded in G provided every Sylow subgroup of H is a Sylow subgroup of some π-quasinormal subgroup of G; H is called c-supplemented in G if there exists a subgroup N of G such that G = HN and H ∩ N ⩽ H _G = Core _G (H). In this paper, finite groups G satisfying the condition that some kinds of subgroups of G are either π-quasinormally embedded or c-supplemented in G, are investigated, and theorems which unify some recent results are given.      

On one class of modules that are close to Noetherian

We consider an R G-module A over a commutative Noetherian ring R. Let G be a group having infinite section p-rank (or infinite 0-rank) such that C _G (A) = 1, A/C _A (G) is not a Noetherian R-module, but the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (or infinite 0-rank, respectively). In this paper, it is proved that if G is a locally soluble group, then G is soluble. Some properties of soluble groups of this type are also obtained.     

A Note on c-Supplemented Subgroups of Finite Groups

A subgroup H of a finite group G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is contained in H _G , where H _G is the largest normal subgroup of G contained in H. In this article, we prove that G is solvable if every subgroup of prime odd order of G is c-supplemented in G. Moreover, we prove that G is solvable if and only if every Sylow subgroup of odd order of G is c-supplemented in G. These results improve and extend the classical results of Hall's articles of (1937) and the recent results of Ballester-Bolinches and Guo's article of (1999), Ballester-Bolinches et al.'s article of (2000), and Asaad and Ramadan's article of (2008).     

On normal subgroups in the periodic products of S.I. Adian

A subgroup H of a given group G is called a hereditarily factorizable subgroup (HF subgroup) if each congruence on H can be extended to some congruence on the entire group G. An arbitrary group G ₁ is an HF subgroup of the direct product G ₁ × G ₂, as well as of the free product G ₁ * G ₂ of groups G ₁ and G ₂. In this paper a necessary and sufficient condition is obtained for a factor Gi of Adian’s n-periodic product Π _i∈I ⁿ G _i of an arbitrary family of groups {G _i } _i∈I to be an HF subgroup. We also prove that for each odd n ≥ 1003 any noncyclic subgroup of the free Burnside group B(m, n) contains an HF subgroup isomorphic to the group B(∞, n) of infinite rank. This strengthens the recent results of A.Yu. Ol’shanskii and M. Sapir, D. Sonkin, and S. Ivanov on HF subgroups of free Burnside groups. This result implies, in particular, that each (noncyclic) subgroup of the group B(m, n) is SQ-universal in the class of all groups of period n. Moreover, it turns out that any countable group of period n is embedded in some 2-generated group of period n, which strengthens the previously obtained result of V. Obraztsov. At the end of the paper we prove that the group B(m, n) is distinguished as a direct factor in any n-periodic group in which it is contained as a normal subgroup.     

Elliptic elements in Möbius groups

Let G be a nonelementary subgroup of Isom(H ⁿ). In this paper we prove that if G contains elliptic elements and dim(M _G ) is even, then G is discrete if and only if WY (G) is discrete and every nonelementary subgroup of G generated by two elliptic elements is discrete. We also describe an example to show the assumption on dim(M _G ) is necessary. Supported in part by NSFC 10671059 and by Leading academic Discipline program, 211 project for Shanghai University of Finance and Economics (the 3rd phase).     

Extendible elements of the alternating groups

An element?σ?of A_n , the Alternating group of degree n, is extendible in S_n , the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

Free products of kω-topological groups with normal amalgamation

It is shown in this paper that if A is a closed normal subgroup of k_ω-topological groups G and H, then the free product of G and H with A amalgamated, G1_AH, exists, is Hausdorff and indeed a k_ω-group.     

On the n-minimal subgroups

Let G be a finite group and n be a positive integer. An n-minimal subgroup H of G is called to be exactly n-minimal if no proper subgroup of H is n-minimal. In this paper, we study the solvability of G under the assumption that all exactly n-minimal subgroups of G are S-permutable.     

Matrix Near-rings over Centralizer Near-rings

For a finite group G and a subgroup A of Aut(G), let M_A(G) denote the centralizer near-ring determined by A and G. The group G is an M_A(G)-module. Using the action of M_A(G) on G, one has the n × n generalized matrix near-ring Mat_n(M_A(G);G). The correspondence between the ideals of M_A(G) and those of Mat_n(M_A(G);G) is investigated. It is shown that if every ideal of M_A(G) is an annihilator ideal, then there is a bijection between the ideals of M_A(G) and those of Mat_n(M_A(G);G).1991 Mathematics Subject Classification: 16Y30     

On g-s-supplemented subgroups of finite groups

A subgroup H of a group G is said to be g-s-supplemented in G if there exists a subgroup K of G such that HK⊴G and H ∩ K ⩽ H _sG , where H_sG is the largest s-permutable subgroup of G contained in H. By using this new concept, we establish some new criteria for a group G to be soluble.     

The influence of <Emphasis Type="Italic">M</Emphasis>-supplemented subgroups on the structure of finite groups

A subgroup H of a group G is said to be M-supplemented in G if there exists a subgroup B of G such that G = HB and T B < G for every maximal sub-group T of H. Moreover, a subgroup H is called c-supplemented in G if there exists a subgroup K such that G = HK and H ∩ K ≤ H _G where H _G is the largest normal subgroup of G contained in H. In this paper we give some conditions of supersolv-ability of finite group under assumption that some primary subgroups have some kinds of supplements, which are generalizations of some recent results.     

A Note on Almost Regular Group Rings

It is proved that a group ring R = AG is almost regular if and only if (i) the ring A is almost regular; (ii) the group G is locally finite; (iii) the order |H| of every finite subgroup H of G is invertible in A. Bibliography: 7 titles.