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We carry out a study of groups G in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite restricted-finite abelian group is isomorphic to Z×K or Zp∞ ×K, where K is a finite group and p is a prime number. We also prove that a group G is infinitely generated restricted-finite if and only if G = AT where A and T are subgroups of G such that A is normal quasi-cyclic and T is finite. As an application of our results, we show that if G is not torsion with finite G′ and the group-ring RG has restricted minimum condition, then R is a semisimple ring and G ∼= T ⋊ Z, where T is finite whose order is unit in R. The converse is also true with certain conditions including G = T × Z. [ABSTRACT FROM AUTHOR]

The neutrosophic automorphisms of a neutrosophic groups G (I) , denoted by Aut(G (I)) is a neu-trosophic group under the usual mapping composition. It is a permutation of G (I) which is also a neutrosophic homomorphism. Moreover, suppose that X1 = X(G (I)) is the neutrosophic group of inner neutrosophic auto-morphisms of a neutrosophic group G (I) and Xn the neutrosophic group of inner neutrosophic automorphisms of Xn-1. In this paper, we show that if any neutrosophic group of the sequence G (I), X1, X2, … is the identity, then G (I) is nilpotent.

The structure of groups which are rich in subnormal subgroups has been investigated by several authors. Here, we prove that if a periodic soluble group G has subnormal deviation, which means that the set of its non-subnormal subgroups satisfies a very weak chain condition, then either G is a Černikov group or all its subgroups are subnormal. It follows that if a periodic soluble group has a subnormal deviation, then its subnormal deviation is 0. [ABSTRACT FROM AUTHOR]

The neutrosophic automorphisms of a neutrosophic groups G (I), denoted by Aut(G (I)) is a neu-trosophic group under the usual mapping composition. It is a permutation of G (I) which is also a neutrosophic homomorphism. Moreover, suppose that X1 = X(G (I)) is the neutrosophic group of inner neutrosophic auto-morphisms of a neutrosophic group G (I) and Xn the neutrosophic group of inner neutrosophic automorphisms of Xn-1. In this paper, we show that if any neutrosophic group of the sequence G (I), X1, X2, ... is the identity, then G (I) is nilpotent. [ABSTRACT FROM AUTHOR]

It is proved that if G is an X -group of infinite rank whose proper subgroups of infinite rank are Baer groups, then so are all proper subgroups of G, where X is the class defined by N.S. Černikov as the closure of the class of periodic locally graded groups by the closure operations P ´ , P ' and L . We prove also that if a locally graded group, which is neither Baer nor Černikov, satisfies the minimal condition on non-Baer subgroups, then it is a Baer-by-Černikov group which is a direct product of a p-subgroup containing a minimal non-Baer subgroup of infinite rank, by a Černikov nilpotent p ′ -subgroup, for some prime p. Our last result states that a group is locally graded and has only finitely many conjugacy classes of non-Baer subgroups if, and only if, it is Baer-by-finite and has only finitely many non-Baer subgroups. [ABSTRACT FROM AUTHOR]

Groups in which every non-abelian subgroup is equal to its normalizer have been recently completely described. This paper investigates locally soluble groups with restrictions on subgroups which are neither abelian nor self-normalizing. [ABSTRACT FROM AUTHOR]

The structure of groups which are rich in subnormal subgroups has been investigated by several authors. Here, we prove that if a periodic soluble group G has subnormal deviation, which means that the set of its non-subnormal subgroups satisfies a very weak chain condition, then either G is a Černikov group or all its subgroups are subnormal. It follows that if a periodic soluble group has a subnormal deviation, then its subnormal deviation is 0.

In this paper, locally graded group satisfying the minimal condition on subgroups which are neither modular nor self-normalizing are described; locally (soluble-by-finite) groups of infinite rank in which every subgroup is either modular or self-normalizing are also characterized in terms of their subgroups of infinite rank. Moreover, the results obtained are used to study groups satisfying similar restrictions on subgroups which are neither permutable nor self-normalizing. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called commensurable with a normal subgroup (cn) if there is N C G such that |HN/(H ∩ N)| is finite. We characterize generalized radical groups G which have one of the following finiteness conditions: (A) the minimal condition on non-cn subgroups of G; (B) the non-cn subgroups of G fall into finitely many conjugacy classes; (C) the non-cn subgroups of G have finite rank.

A subgroup H of a group G is said to be permutable if H K = K H for every subgroup K of G and the group G is called metaquasihamiltonian if all subgroups of G are either permutable or abelian. It is known that a locally graded metaquasihamiltonian group G is soluble with derived length at most 4 and contains a finite normal subgroup N such that all subgroups of the factor G / N are permutable. In this paper, we describe locally graded groups in which the set of all nonmetaquasihamiltonian subgroups satisfies the minimal condition and locally graded groups with the minimal condition on subgroups which are neither abelian nor permutable. Moreover, it is proved here that a finitely generated hyper-(abelian or finite) group whose finite homomorphic images are metaquasihamiltonian is itself metaquasihamiltonian. [ABSTRACT FROM AUTHOR]

It is proved here that if G is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then G is either locally supersoluble or a \vCernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any infinite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble.

Let c be a positive integer. A group G is called an FC-group if each element of G has only finitely many conjugates by G, and G lies in the FC-center of G. The FC-groups with the minimal condition or the maximal conditions on abelian subgroups are investigated and some characterizations of them are obtained. A group is called an FC- soluble group if it possesses an FC-series of finite length. Another aim of this article is to give necessary and sufficient conditions for FC-soluble groups to satisfy the minimal condition or the maximal conditions on abelian subgroups. [ABSTRACT FROM AUTHOR]

It is proved here that if G is a locally graded group satisfying the minimal condition on subgroups which are not locally supersoluble, then G is either locally supersoluble or a Cernikov group. The same conclusion holds for locally finite groups satisfying the weak minimal condition on non-(locally supersoluble) subgroups. As a consequence, it is shown that any in nite locally graded group whose non-(locally supersoluble) subgroups lie into finitely many conjugacy classes must be locally supersoluble [ABSTRACT FROM AUTHOR]

Given an element a of a left brace A satisfying some nilpotency conditions, we describe the smallest subbrace of A containing a. We also present a description of the left braces satisfying the minimal condition for subbraces. [ABSTRACT FROM AUTHOR]

It is known that there exist soluble groups of infinite rank which satisfy the minimal condition on normal subgroups. We prove here that if G is any soluble group satisfying the minimal condition on normal subgroups of infinite rank, then either G has finite rank or it satisfies the minimal condition on normal subgroups. [ABSTRACT FROM AUTHOR]

We establish several characterizations of maximal non-Prüfer and maximal non-integrally closed subrings of a field. Special attention is given when finiteness conditions are satisfied. Several numerical characterizations are then obtained. The paper is concluded with examples that exhibit the obtained results. [ABSTRACT FROM AUTHOR]

A ring extension R ⊆ S is said to be FO if it has only finitely many intermediate rings. R ⊆ S is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension R ⊆ S to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair with only finitely many intermediate rings. We also obtain as a corollary several new and old characterizations of Prüfer and integral domains satisfying the corresponding finiteness conditions. [ABSTRACT FROM AUTHOR]

In this paper, the structure of (generalized) soluble groups for which the set of all subnormal non-normal subgroups satisfies the minimal condition is described, taking as a model the known theory of groups in which normality is a transitive relation. [ABSTRACT FROM AUTHOR]

We characterize the solvable groups satisfying the maximal condition and the minimal condition on subgroups of infinite exponent, respectively. [ABSTRACT FROM AUTHOR]

We study the structure of groups that satisfy the maximal conditions on infinite normal subgroups and the minimal condition on normal subgroups of infinite index. Characterizations of both properties are given in the general case. In addition, detailed information is obtained about the structure of soluble groups that satisfy these chain conditions. [ABSTRACT FROM AUTHOR]

The authors study infinite dimensional linear groups with the minimal condition on subgroups of infinite central dimension. [Copyright &y& Elsevier]