On Groups with Chain Conditions

1971 ◽  
Vol 23 (1) ◽  
pp. 151-159
Author(s):  
Bernhard Amberg
Keyword(s):  
Minimum Condition ◽  
Normal Subgroups ◽  
Maximum Condition ◽  
Finiteness Conditions ◽  
Chain Conditions ◽  
Radical Group ◽  
Abelian Subgroups ◽  
Maximum Minimum ◽  
Subnormal Subgroups

Our aim in this note is to generalize results of Baer in [3; 5]. In § 1 an arbitrary formation n is considered, the key result being Proposition 1.5. This is applied in § 2 to characterize various finiteness conditions, for example the classes of groups with maximum [minimum] condition on subgroups, subnormal subgroups, and normal subgroups respectively, or the class of (not necessarily soluble) polyminimax groups (see Theorems 2.1 and 2.6). These results may also be regarded as generalizations of the well-known theorem of Malcev-Baer that a radical group satisfies the maximum condition [is a polyminimax group] if all its abelian subgroups satisfy the maximum condition [are minimax groups].

scholarly journals Rings whose additive endomorphisms are ring endomorphisms

1990 ◽  
Vol 42 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Gary Birkenmeier ◽  
Henry Heatherly
Keyword(s):  
Subdirectly Irreducible ◽  
Finiteness Conditions ◽  
Chain Conditions ◽  
Open Questions ◽  
Open Case ◽  
Ring Endomorphism ◽  
Substantial Progress ◽  
Additive Endomorphism ◽  
Made In

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

A NOTE ON GROUPS WHOSE PROPER LARGE SUBGROUPS HAVE A TRANSITIVE NORMALITY RELATION

2016 ◽  
Vol 95 (1) ◽  
pp. 38-47 ◽  
Author(s):  
FRANCESCO DE GIOVANNI ◽  
MARCO TROMBETTI
Keyword(s):  
Soluble Group ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Transitive Relation ◽  
Formula Group ◽  
Subnormal Subgroups

A group $G$ is said to have the $T$-property (or to be a $T$-group) if all its subnormal subgroups are normal, that is, if normality in $G$ is a transitive relation. The aim of this paper is to investigate the behaviour of uncountable groups of cardinality $\aleph$ whose proper subgroups of cardinality $\aleph$ have a transitive normality relation. It is proved that such a group $G$ is a $T$-group (and all its subgroups have the same property) provided that $G$ has an ascending subnormal series with abelian factors. Moreover, it is shown that if $G$ is an uncountable soluble group of cardinality $\aleph$ whose proper normal subgroups of cardinality $\aleph$ have the $T$-property, then every subnormal subgroup of $G$ has only finitely many conjugates.

SOME CRITERIA FOR EXISTENCE OF SUPPLEMENTS TO NORMAL SUBGROUPS AND THEIR APPLICATIONS

2010 ◽  
Vol 20 (05) ◽  
pp. 689-719 ◽  
Author(s):  
LEONID A. KURDACHENKO ◽  
JAVIER OTAL ◽  
IGOR YA. SUBBOTIN
Keyword(s):  
Normal Subgroups ◽  
Finitely Generated ◽  
Original Proof ◽  
Soluble Groups ◽  
Finite Section ◽  
Abelian Subgroups ◽  
Section Rank ◽  
The Many ◽  
New Criteria

We established several new criteria for existence of complements and supplements to some normal abelian subgroups in groups. In passing, as one of the many useful applications and corollaries of these results, we obtained a description of some finitely generated soluble groups of finite Hirsch–Zaitsev rank. As another application of our results, we obtained a D.J.S. Robinson's theorem on structure of finitely generated soluble groups of finite section rank. The original proof of this theorem was homological, but all proofs in this paper, including this one, are purely group-theoretical.

Groups with few conjugacy classes of non-normal subgroups

2008 ◽  
Vol 58 (2) ◽  
Author(s):  
Maria Falco ◽  
Francesco Giovanni ◽  
Carmela Musella
Keyword(s):  
Abelian Subgroup ◽  
Soluble Group ◽  
Commutator Subgroup ◽  
Conjugacy Classes ◽  
Normal Subgroups ◽  
Abelian Subgroups ◽  
Locally Soluble Group

AbstractThe structure of groups with finitely many non-normal subgroups is well known. In this paper, groups are investigated with finitely many conjugacy classes of non-normal subgroups with a given property. In particular, it is proved that a locally soluble group with finitely many non-trivial conjugacy classes of non-abelian subgroups has finite commutator subgroup. This result generalizes a theorem by Romalis and Sesekin on groups in which every non-abelian subgroup is normal.

On Groups Saturated with Abelian Subgroups

1998 ◽  
Vol 08 (04) ◽  
pp. 443-466 ◽  
Author(s):  
Lev S. Kazarin ◽  
Leonid A. Kurdachenko ◽  
Igor Ya. Subbotin
Keyword(s):  
Finite Groups ◽  
Abelian Groups ◽  
Normal Subgroups ◽  
Main Subject ◽  
Maximal Condition ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Locally Nilpotent ◽  
Abelian Subgroups ◽  
Weak Maximal Condition

Groups with the weak maximal condition on non-abelian subgroups are the main subject of this research. Locally finite groups with this property are abelian or Chemikov. Non-abelian groups with the weak maximal condition on non-abelian subgroups, which have an ascending series of normal subgroups with locally nilpotent or locally finite factors, are described in this article.

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