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de Giovanni, Francesco; Trombetti, Marco

A note on groups whose proper large subgroups have a transitive normality relation. (English) Zbl 1377.20020

Bull. Aust. Math. Soc. 95, No. 1, 38-47 (2017).
In the article under review, the authors investigate the structure of uncountable groups whose proper subgroups of cardinality \(\aleph\), \(\aleph\) an uncountable cardinal, have a transitive normality relation or the \(T\)-property for shortness. They proved the following two main results.
1.
{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.}
Recall that a group is subsoluble if it has an ascending series with abelian factors consisting of subnormal subgroups. Then, the second result is the following.
2.
{If \(G\) is an uncountable subsoluble group of cardinality \(\aleph\) whose proper subgroups of cardinality \(\aleph\) have the \(T\)-property then all its subgroups also have the \(T\)-property.}
Reviewer: Bui Xuan Hai (Ho Chi Minh City)

Cited in 2 Documents

MSC:

20E15 Chains and lattices of subgroups, subnormal subgroups
20F19 Generalizations of solvable and nilpotent groups
20F16 Solvable groups, supersolvable groups
20E07 Subgroup theorems; subgroup growth
20E34 General structure theorems for groups

Keywords:

uncountable groups; T-property; T-group
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References:

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