Locally soluble groups with all nontrivial normal subgroups isomorphic.

John C. Lennox; Howard Smith; James Wiegold

Displaying similar documents to “Locally soluble groups with all nontrivial normal subgroups isomorphic.”

Groups whose proper subgroups are Baer-by-Chernikov or Baer-by-(finite rank)

Abdelhafid Badis, Nadir Trabelsi (2011)

Open Mathematics

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Our main result is that a locally graded group whose proper subgroups are Baer-by-Chernikov is itself Baer-by-Chernikov. We prove also that a locally (soluble-by-finite) group whose proper subgroups are Baer-by-(finite rank) is itself Baer-by-(finite rank) if either it is locally of finite rank but not locally finite or it has no infinite simple images.

Groups with all subgroups permutable or of finite rank

Martyn Dixon, Yalcin Karatas (2012)

Open Mathematics

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In this paper we investigate the structure of X-groups in which every subgroup is permutable or of finite rank. We show that every subgroup of such a group is permutable.

A note on groups of finite rank

Derek J. S. Robinson (1969)

Compositio Mathematica

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Groups with the weak minimal condition for non-subnormal subgroups II

Leonid A. Kurdachenko, Howard Smith (2005)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a group with the property that there are no infinite descending chains of non-subnormal subgroups of $G$ for which all successive indices are infinite. The main result is that if $G$ is a locally (soluble-by-finite) group with this property then either $G$ has subgroups subnormal or $G$ is a soluble-by-finite minimax group. This result fills a gap left in an earlier paper by the same authors on groups with the stated property.

A note on groups of infinite rank whose proper subgroups are abelian-by-finite

Francesco de Giovanni, Federica Saccomanno (2014)

Colloquium Mathematicae

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It is proved that if G is a locally (soluble-by-finite) group of infinite rank in which every proper subgroup of infinite rank contains an abelian subgroup of finite index, then all proper subgroups of G are abelian-by-finite.

Soluble products of two locally finite groups with min- $p$ for every prime $p$

Bernhard Amberg (1983)

Rendiconti del Seminario Matematico della Università di Padova

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Groups with finitely many conjugacy classes of non-normal subgroups of infinite rank

Maria De Falco, Francesco de Giovanni, Carmela Musella (2013)

Colloquium Mathematicae

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It is proved that if a locally soluble group of infinite rank has only finitely many non-trivial conjugacy classes of subgroups of infinite rank, then all its subgroups are normal.

Groups with small deviation for non-subnormal subgroups

Leonid Kurdachenko, Howard Smith (2009)

Open Mathematics

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We introduce the notion of the non-subnormal deviation of a group G. If the deviation is 0 then G satisfies the minimal condition for nonsubnormal subgroups, while if the deviation is at most 1 then G satisfies the so-called weak minimal condition for such subgroups (though the converse does not hold). Here we present some results on groups G that are either soluble or locally nilpotent and that have deviation at most 1. For example, a torsion-free locally nilpotent with deviation at...

Anti-CC-groups and anti-PC-groups.

Russo, Francesco (2007)

International Journal of Mathematics and Mathematical Sciences

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The nilpotency of some groups with all subgroups subnormal.

Leonid A. Kurdachenko, Howard Smith (1998)

Publicacions Matemàtiques

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Let G be a group with all subgroups subnormal. A normal subgroup N of G is said to be G-minimax if it has a finite G-invariant series whose factors are abelian and satisfy either max-G or min- G. It is proved that if the normal closure of every element of G is G-minimax then G is nilpotent and the normal closure of every element is minimax. Further results of this type are also obtained.