Groups with weak maximality condition for nonnilpotent subgroups

Abstract

A group G satisfies the weak maximality condition for nonnilpotent subgroups [or, briefly, the Wmax-(nonnil) condition if G does not have infinite increasing chains {Hn | n ∈ N} of nonnilpotent subgroups such that the indices |Hn+1 : Hn| are infinite for each n ∈ N. We study the structure of hypercentral groups satisfying the weak maximality condition for nonnilpotent subgroups.Let L nonnil (G) be a family of all nonnilpotent subgroups of a group G. The study of groups in which the family L nonnil (G) is "very small" (in a certain sense) was originated by the classical work by Shmidt [1] devoted to the investigation of finite groups all proper subgroups of which are nilpotent (i.e., {G} = L nonnil (G)) . If G is an infinite group all proper subgroups of which are nilpotent, then G is either finitely generated or locally nilpotent. For the first time, these groups were studied by Newman and Wiegold in [2]. Note that the examples of infinite finitely generated groups all proper subgroups of which are Abelian constructed by Ol'shanskii [3] (Sec. 28) demonstrate that, at present, the possibility of investigation of infinite finitely generated groups all proper subgroups of which are nilpotent seems to be problematic. We also note that a series of examples of infinitely generated groups all proper subgroups of which are nilpotent is presented in [4][5][6][7][8]. However, the problem of complete investigation of these groups is also far from being solved. In [9], Smith described the most general features of the structure of solvable groups with the indicated property and obtained the first results for groups in which the family L nonnil (G) satisfies both the maximality and minimality conditions and the weak maximality and minimality conditions. We now recall the general definition of the weak minimality and maximality conditions. Let M be a system of subgroups of the group G. We say that M satisfies a weak maximality (minimality) condition or the group G satisfies the weak maximality (minimality) condition for M -subgroups or, briefly, the Wmax-M ( Wmin-M ) condition if G does not contain infinite increasing (decreasing) chains {H n | n ∈ N} of subgroups from the system M such that the indices |H n+1 : H n | ( |H n : H n+1 | ) are infinite for any n ∈ N. If M is the system of all nonnilpotent groups, then we get groups with weak maximality (minimality) condition for nonnilpotent subgroups or, briefly, groups with Wmax-(nonnil) ( Wmin-(nonnil) ) condition. The notions of weak minimality and maximality conditions were introduced by Zaitsev [10] and Baer [11] and efficiently used in various investigations (see, e.g., the surveys [12,13]). After the cited Smith's work, a fairly detailed investigation of groups in which the family L nonnil (G) satisfies the maximality condition was carried out by Dixon and Kurdachenko [14,15]. Groups in which the family L nonnil (G) satisfies the minimality condition were studied by Dixon, Evans, and Smith [16]. In [17], these authors studied groups in which the family L nonnil (G) satisf...

Locally (Soluble-by-Finite) Groups With Various Restrictions on Subgroups of Infinite Rank

“…Theorem 1 may be viewed as belonging to a larger family of results stating that a group with "many" subgroups having a certain property P, or "few" subgroups not having P, itself satisfies P. Among the many articles on this broad topic, in addition to those referred to above, we mention here [5], [7] and [13]. For a property P of groups, a group G satisfies the weak maximal condition for non-P subgroups if there is no infinite ascending chain H 0 < H 1 < .…”