On the \(n\)-nilpotency (Q1411543)

The main question discussed in the paper under review is some cases of the following: Does an algebra (not necessarily associative) have a property \(\mathcal P\) if it is the sum of two ideals each of which has the property \(\mathcal P\)? If a group is the product of two normal subgroups having a property \(\mathcal P\), does it have itself the property \(\mathcal P\)? Some positive solutions to these problems are known, for example the locally nilpotent and the locally finite cases. Also, the product of two periodic normal subgroups is periodic and the sum of two nil associative ideals is nil. Let \(n\) be a positive integer. A group (resp. algebra) is said to be \(n\)-nilpotent if every \(n\)-generator subgroup (resp. subalgebra) is nilpotent. The problem whether the product of \(n\)-nilpotent normal subgroups is \(n\)-nilpotent was formulated by \textit{Sh. S. Kemkhadze} for \(n=2\) [Problem 1.39 of The Kourovka notebook. Unsolved problems in group theory. 13th augm. ed. Novosibirsk: Institute of Mathematics. Russian Academy of Sciences, Siberian Division (1995; Zbl 0838.20001)] and for every integer \(n\) by \textit{O. H. Kegel} and \textit{B. A. F. Wehrfritz} [Page 2 of Locally finite groups. North-Holland Mathematical Library. Vol. 3. Amsterdam-London: North-Holland Publishing Comp.; New York: American Elsevier Publishing Comp., Inc. (1973; Zbl 0259.20001)]. In [Thèse, Univ. Haute-Alsace (1996); Preprint IRMA (Strasbourg) 1996/08 and Rend. Semin. Mat. Univ. Padova 102, 219-232 (1999; Zbl 1073.16503)] the author asked whether such a product may not be \(2\)-nilpotent. Here he solves the problem in the affirmative giving best possible answers to the question of Kemkhadze and Kegel-Wehrfritz while improving on \textit{A. I. Sozutov}'s and the author's results [see the last two references and Algebra Logic 30, No. 1, 70-72 (1991); translation from Algebra Logika 30, No. 1, 102-105 (1991; Zbl 0820.20031) and Math. Notes 57, No. 3, 307-309 (1995); translation from Mat. Zametki 57, No. 3, 445-450 (1995; Zbl 0840.16016)]. The main result of the paper is: Theorem. For any integers \(n\geq 1\), \(d\geq m\geq 2\) and over any field there exists a residually nilpotent \(d\)-generator non-nilpotent associative nilalgebra which is the sum of \(m\) \(n\)-nilpotent ideals. As corollaries to this result the author proves the following: Corollary 1. For every integer \(n\geq 1\) and over any field there exists a residually nilpotent 2-generator non-nilpotent associative nilalgebra which is the sum of two \(n\)-nilpotent ideals. Corollary 2. For any integers \(n\geq 1\), \(d\geq m\geq 2\) and over any field (resp. any field containing \(1/2\)) there exists a non-\(d\)-nilpotent, residually finite nil Lie (resp. Jordan) algebra which is the sum of \(m\) \(n\)-nilpotent ideals. Corollary 3. For any integers \(n\geq 1\), \(d\geq m\geq 2\) and every prime number \(p\) (resp. \(p=0\)), there exists a non-\(d\)-nilpotent, residually finite (resp. residually nilpotent) \(p\)-group (resp. torsion free group) which is the product of \(m\) \(n\)-nilpotent normal subgroups. Corollary 4. For every integer \(n\geq 1\) and every prime \(p>2\), there exists a residually finite, \(n\)-finite \(p\)-group generated by \(n+1\) conjugate elements of order \(p\).