A group is called metahamiltonian if all its non-abelian subgroups are normal; it is known that locally soluble metahamiltonian groups have finite commutator subgroup. Here the structure of locally graded groups with finitely many normalizers of (infinite) non-abelian subgroups is investigated, and the above result is extended to this more general situation.

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The structure of soluble groups in which normality is a transitive relation is known. Here, groups with finitely many normalizers of subnormal subgroups are investigated, and the behavior of the Wielandt subgroup of such groups is described; moreover, groups having only finitely many normalizers of infinite subnormal subgroups are considered.

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