Groups with some restrictions on non-Baer subgroups


It is proved that if G is an \(\mathfrak {X}\)-group of infinite rank whose proper subgroups of infinite rank are Baer groups, then so are all proper subgroups of G, where \(\mathfrak {X}\) is the class defined by N.S. Černikov as the closure of the class of periodic locally graded groups by the closure operations \(\varvec{\acute{P}}\), \(\varvec{\grave{P}}\) and \( \varvec{L}\). We prove also that if a locally graded group, which is neither Baer nor Černikov, satisfies the minimal condition on non-Baer subgroups, then it is a Baer-by-Černikov group which is a direct product of a p-subgroup containing a minimal non-Baer subgroup of infinite rank, by a Černikov nilpotent \(p^{\prime }\)-subgroup, for some prime p. Our last result states that a group is locally graded and has only finitely many conjugacy classes of non-Baer subgroups if, and only if, it is Baer-by-finite and has only finitely many non-Baer subgroups.

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Both authors are grateful to the referee of a previous version of this work for careful reading and many suggestions improving the presentation of this paper.

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Correspondence to Nadir Trabelsi.

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Badis, A., Trabelsi, N. Groups with some restrictions on non-Baer subgroups. Ricerche mat (2021).

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  • Baer
  • Locally (soluble-by-finite)
  • Minimal condition
  • Rank

Mathematics Subject Classification

  • 20F19
  • 20F99