A Note on Nilpotent-by-Ernikov Groups

Abstract

Abstract. In this note we prove that a locally graded group G in which all proper subgroups are (nilpotent of class not exceeding n)-by-Černikov, is itself (nilpotent of class not exceeding n)-by-Černikov.As a preparatory result that is used for the proof of the former statement in the case of a periodic group, we also prove that a group G, containing a nilpotent of class n subgroup of finite index, also contains a characteristic subgroup of finite index that is nilpotent of class not exceeding n.2000 Mathematics Subject Classification. Primary 20F19. Secondary 20F22.

Finite groups and Lie rings with a metacyclic Frobenius group of automorphisms

“…By a result of B. Bruno and F. Napolitani [1,Lemma 3] if a group has a subgroup of finite index k that is nilpotent of class l, then it also has a characteristic subgroup of finite (k, l)-bounded index that is nilpotent of class l. Therefore in the proof of Theorem 1.1 we only need a subgroup of (m, n, c)-bounded index and of (c, q)-bounded nilpotency class.…”

Groups With Finitely Many Conjugacy Classes of Subgroups That Are Not Nilpotent-by-Chernikov

“…We shall denote by N n C the class of groups under discussion (and by NC the class of nilpotent-by-Chernikov groups). Using this result from [2], our final theorem is a very easy consequence of Theorem 3.…”

“…Suppose that G satisfies the hypotheses but not the conclusion of Theorem 4. By Theorem 3, G is nilpotent-by-Chernikov; in particular G is L(NF), and another application of Lemma 3.2 of [4], using the main result of [2] in place of Theorem 1 above, gives the desired contradiction.…”

“…There is an interesting "bounded version" of Theorem 1, proved in [2]. This states that a locally graded group in which every proper subgroup is (nilpotent of class at most n)-by-Chernikov (n a fixed positive integer) is (nilpotent of class at most n)-by-Chernikov.…”

“…One consequence of Theorem 6 of that paper is that a locally finite p-group with all proper subgroups (nil-n)-by-Chernikov is soluble and nilpotent-by-Chernikov.) We remark that the case n = 1 of the above theorem from [2] was successfully dealt with in [7], subsequent to the proof for G periodic (and locally graded) in [9]. We shall denote by N n C the class of groups under discussion (and by NC the class of nilpotent-by-Chernikov groups).…”