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A NOTE ON GROUPS WHOSE PROPER LARGE SUBGROUPS HAVE A TRANSITIVE NORMALITY RELATION
Published online by Cambridge University Press: 04 November 2016
Abstract
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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- Research Article
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- © 2016 Australian Mathematical Publishing Association Inc.
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