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**Locally graded groups with certain minimal conditions for subgroups. II.**
*(English)*
Zbl 0662.20028

[For Part I cf. the preceding item Zbl 0662.20027.]

These two papers deal with one of the ways of studying infinite groups many of whose subgroups have a prescribed property, namely the consideration of minimal conditions in the direction described by R. E. Phillips and J. Wilson [J. Algebra 51, 41-68 (1978; Zbl 0374.20042)]. The second paper extends the main result of the first one, and these results are the following.

Theorem. Let P be one of the following properties: (i) abelian; (ii) normal; (iii) normal or abelian; (iv) metahamiltonian. If G is a locally graded group which satisfies the minimal condition for subgroups not having P then either G is a Chernikov group or every subgroup of G satisfies P. - Theorem. Let c be a nonnegative integer and let P be one of the following properties: (i) normal; (ii) locally nilpotent; (iii) normal or nilpotent of class at most c; (iv) normal or locally nilpotent; (v) normal or nilpotent of class at most c; (vi) c-Hamiltonian (that is, a group G is said to be c-Hamiltonian if every subgroup of G is either normal in G or nilpotent of class at most c). If G is a locally graded group, then we have: 1) A minimal non-P subgroup of G is finite. 2) G satisfies the minimal condition for subgroups not having P if and only if either G is a Chernikov group or every subgroup of G satisfies P.

These two papers deal with one of the ways of studying infinite groups many of whose subgroups have a prescribed property, namely the consideration of minimal conditions in the direction described by R. E. Phillips and J. Wilson [J. Algebra 51, 41-68 (1978; Zbl 0374.20042)]. The second paper extends the main result of the first one, and these results are the following.

Theorem. Let P be one of the following properties: (i) abelian; (ii) normal; (iii) normal or abelian; (iv) metahamiltonian. If G is a locally graded group which satisfies the minimal condition for subgroups not having P then either G is a Chernikov group or every subgroup of G satisfies P. - Theorem. Let c be a nonnegative integer and let P be one of the following properties: (i) normal; (ii) locally nilpotent; (iii) normal or nilpotent of class at most c; (iv) normal or locally nilpotent; (v) normal or nilpotent of class at most c; (vi) c-Hamiltonian (that is, a group G is said to be c-Hamiltonian if every subgroup of G is either normal in G or nilpotent of class at most c). If G is a locally graded group, then we have: 1) A minimal non-P subgroup of G is finite. 2) G satisfies the minimal condition for subgroups not having P if and only if either G is a Chernikov group or every subgroup of G satisfies P.

Reviewer: J.Otal

### MSC:

20F22 | Other classes of groups defined by subgroup chains |

20E15 | Chains and lattices of subgroups, subnormal subgroups |