Uncountable groups in which normality is a transitive relation

A group is called a [Formula: see text]-group if all its subnormal subgroups are normal. It is proved here that if [Formula: see text] is an uncountable soluble periodic group of regular cardinality [Formula: see text] in which all subnormal subgroups of cardinality [Formula: see text] are normal, then [Formula: see text] is a [Formula: see text]-group, provided that it satisfies a suitable residual condition. An example shows that this latter restriction cannot be avoided.

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

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972716000848 ◽

2016 ◽

Vol 95 (1) ◽

pp. 38-47 ◽

Cited By ~ 1

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.

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ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS

Glasgow Mathematical Journal ◽

10.1017/s0017089514000159 ◽

2014 ◽

Vol 56 (3) ◽

pp. 691-703 ◽

Cited By ~ 1

Author(s):

A. BALLESTER-BOLINCHES ◽

J. C. BEIDLEMAN ◽

A. D. FELDMAN ◽

M. F. RAGLAND

Keyword(s):

Finite Group ◽

Finite Groups ◽

Subnormal Subgroup ◽

Normal Subgroups ◽

Transitive Relation ◽

Formula Group ◽

Permutable Subgroups ◽

Sylow Permutable ◽

A Finite Group ◽

Subnormal Subgroups

AbstractFor a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

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GROUPS OF INFINITE RANK IN WHICH NORMALITY IS A TRANSITIVE RELATION

Glasgow Mathematical Journal ◽

10.1017/s0017089513000323 ◽

2013 ◽

Vol 56 (2) ◽

pp. 387-393 ◽

Cited By ~ 8

Author(s):

M. DE FALCO ◽

F. DE GIOVANNI ◽

C. MUSELLA ◽

Y. P. SYSAK

Keyword(s):

Finite Rank ◽

Soluble Group ◽

Transitive Relation ◽

Fitting Subgroup ◽

Arbitrary Group ◽

Infinite Rank ◽

Subnormal Subgroups

AbstractA group is called a T-group if all its subnormal subgroups are normal. It is proved here that if G is a periodic (generalized) soluble group in which all subnormal subgroups of infinite rank are normal, then either G is a T-group or it has finite rank. It follows that if G is an arbitrary group whose Fitting subgroup has infinite rank, then G has the property T if and only if all its subnormal subgroups of infinite rank are normal.

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