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- W48242770 abstract "We obtain a characterization of solvable groups with the minimal condi- tion on non-hypercentral (respectively non-nilpotent) subgroups. Mathematics subject classification: 20F16, 20F19, 20F22. Keywords and phrases: Hypercentral group, nilpotent group, HM � -group, minimal condition. Introduction. We say that a group G satisfies the minimal condition on non-hypercentral subgroups Min-ZA (respectively, the minimal condition on non- nilpotent subgroups Min-N) if for any properly descending chain G1 � G2 � ··· � Gn � ··· of subgroups Gn in G there exists a number m 2 N such that Gn is hy- percentral (respectively, nilpotent) for each nm. The minimal non-hypercentral (respectively, non-nilpotent) group satisfies Min-ZA (respectively, Min-N). Recall ifis a property pertaining to subgroups and an infinite group G is not a �-group but all its proper subgroups have �, then G is called a minimal non-� group. Any Heineken-Mohamed type group (i.e. non-nilpotent groups with nilpotent and sub- normal proper subgroups) is a minimal non-nilpotent group (1) (see e.g. (2)) and satisfies Min-N and Min-ZA. In this paper we study the groups that satisfy Min-ZA (respectively Min-N) and prove the following Theorem. Let G be a solvable group. Then G satisfies the minimal condition on non-hypercentral (respectively, non-nilpotent) subgroups if and only if one of the following holds: (1) G is a hypercentral (respectively, nilpotent) group; (2) G is a y Cernikov group; (3) G = P ×Q is a group direct product of a hypercentral (respectively, nilpotent) y Cernikov p ' -group Q and a non-hypercentral (respectively, non-nilpotent) p-group P which contains a normal HM ∗ -subgroup H of finite index (respectively, HM ∗ - subgroup H of finite index with the nilpotent commutator subgroup H ' ) with the normalizer condition." @default.
- W48242770 created "2016-06-24" @default.
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- W48242770 date "2008-01-01" @default.
- W48242770 modified "2023-09-27" @default.
- W48242770 title "Groups with many hypercentral subgroups" @default.
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