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2013•Glasgow Mathematical JournalOpen access

GROUPS OF INFINITE RANK IN WHICH NORMALITY IS A TRANSITIVE RELATION

Maria De Falco, Francesco de Giovanni, Carmela Musella, Yaroslav P. Sysak

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Abstract

Abstract A group is called aT-group if all its subnormal subgroups are normal. It is proved here that ifGis a periodic (generalized) soluble group in which all subnormal subgroups of infinite rank are normal, then eitherGis aT-group or it has finite rank. It follows that ifGis an arbitrary group whose Fitting subgroup has infinite rank, thenGhas the propertyTif and only if all its subnormal subgroups of infinite rank are normal.

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Abstract A group is called aT-group if all its subnormal subgroups are normal. It is proved here that ifGis a periodic (generalized) soluble group in which all subnormal subgroups of infinite rank are normal, then eitherGis aT-group or it has finite rank. It follows that ifGis an arbitrary group whose Fitting subgroup has infinite rank, thenGhas the propertyTif and only if all its subnormal subgroups of infinite rank are normal.

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Abstract A group is called aT-group if all its subnormal subgroups are normal. It is proved here that ifGis a periodic (generalized) soluble group in which all subnormal subgroups of infinite rank are normal, then eitherGis aT-group or it has finite rank. It follows that ifGis an arbitrary group whose Fitting subgroup has infinite rank, thenGhas the propertyTif and only if all its subnormal subgroups of infinite rank are normal.

Key concepts: Mathematics, Rank (graph theory), Normal subgroup, Combinatorics, Group (periodic table), Normality, Transitive relation, Pure mathematics

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