Let G be a group and P be a property of groups. If every proper subgroup of G satisfies P but G itself does not satisfy it, then G is called a minimal non-P group. In this work we study locally nilpotent minimal non-P groups, where P stands for ' hypercentral ' or ' nilpotent-by-Chernikov '. In the first case we show that if G is a minimal non-hypercentral Fitting group in which every proper subgroup is solvable, then G is solvable (see Theorem 1.1 below). This result generalizes [3, Theorem 1]. In the second case we show that if every proper subgroup of G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov (see Theorem 1.3 below). This settles a question which was considered in [1-3, 10]. Recently in [9], the non-periodic case of the above question has been settled but the same work contains an assertion without proof about the periodic case.The main results of this paper are given below (see also [13]). T 1.1. Let G be a minimal non-hypercentral Fitting p-group. If e ery proper subgroup of G is sol able, then G is sol able. Now, as an easy corollary to this theorem, we can state the following. C 1.2 [3, Theorem 1]. Let G be a barely transiti e locally nilpotent pgroup. If a point stabilizer in G is sol able and hypercentral, then Gh G. T 1.3. Let G be a locally nilpotent p-group in which e ery proper subgroup is nilpotent-by-Cherniko . Then G is nilpotent-by-Cherniko .As a special case of Theorem 1.3, we can state the following, which answers a question of [5]. C 1.4. Let G be a locally nilpotent minimal non-(nilpotent-by-finite) pgroup. Then G cannot be perfect and hence G\Gh % C p _.The notations and the definitions are standard and may be found in [6] and [11], but we state here those that are frequently used. Let X, Y be nonempty subsets of a

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Abstract. Totally imprimitive p-groups satisfying the cyclic-block property are investigated. It is shown that in these groups any two blocks either are disjoint or one is contained in the other, and so the set of all blocks of the same size forms just one block system. Furthermore the non-FC-subgroups of these groups are transitive. For each prime p totally imprimitive p-subgroups of FSymðN Ã Þ satisfying the cyclic-block property are constructed, which are not minimal non-FC-groups.

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