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Unsolved Problems in Group Theory. the Kourovka Notebook

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Unsolved problems in group theory. The Kourovka notebook. No. 18 Edited by: Mazurov, V. D. and Khukhro, E. I. 2014 MIMS EPrint: 2014.1 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 Russian Academy of Sciences Siberian Division q INSTITUTEOFMATHEMATICS UNSOLVED PROBLEMS IN GROUP THEORY THE KOUROVKA NOTEBOOK No. 18 Novosibirsk 2014 q Editors: V. D. Mazurov and E. I. Khukhro New problems and comments can be sent to the Editors: V. D. Mazurov or E. I. Khukhro Institute of Mathematics Novosibirsk-90, 630090, Russia e-mails: [email protected] [email protected] c V. D. Mazurov, E. I. Khukhro, 2014 Contents Preface ......................................................................... 4 Problems from the 1st Issue (1965) . 5 Problems from the 2nd Issue (1966) . 7 Problems from the 3rd Issue (1969) . 10 Problems from the 4th Issue (1973) . 12 Problems from the 5th Issue (1976) . 16 Problems from the 6th Issue (1978) . 19 Problems from the 7th Issue (1980) . 23 Problems from the 8th Issue (1982) . 27 Problems from the 9th Issue (1984) . 33 Problems from the 10th Issue (1986) . 39 Problems from the 11th Issue (1990) . 47 Problems from the 12th Issue (1992) . 59 Problems from the 13th Issue (1995) . 67 Problems from the 14th Issue (1999) . 73 Problems from the 15th Issue (2002) . 85 Problems from the 16th Issue (2006) . 98 Problems from the 17th Issue (2010) . 111 New Problems . 126 Archive of Solved Problems . 141 Index of Names . 213 4 Preface The idea of publishing a collection of unsolved problems in Group Theory was proposed by M. I. Kargapolov (1928{1976) at the Problem Day of the First All{Union (All{USSR) Symposium on Group Theory which took place in Kourovka, a small village near Sverdlovsk, on February, 16, 1965. This is why this collection acquired the name \Kourovka Notebook". Since then every 2{4 years a new issue has appeared containing new problems and incorporating the problems from the previous issues with brief comments on the solved problems. For more than 40 years the \Kourovka Notebook" has served as a unique means of communication for researchers in Group Theory and nearby fields of mathematics. Maybe the most striking illustration of its success is the fact that more than 3/4 of the problems from the first issue have now been solved. Having acquired international popularity the \Kourovka Notebook" includes problems by more than 300 authors from all over the world. Starting from the 12th issue it is published simultaneously in Russian and in English. This is the 18th issue of the \Kourovka Notebook". It contains 120 new prob- lems. Comments have been added to those problems from the previous issues that have been recently solved. Some problems and comments from the previous issues had to be altered or corrected. The Editors thank all those who sent their remarks on the previous issues and helped in the preparation of the new issue. We thank A. N. Ryaskin for assistance in dealing with the electronic publication. The section \Archive of Solved Problems" contains all solved problems that have already been commented on in one of the previous issues with a reference to a detailed publication containing a complete answer. However, those problems that are com- mented on with a complete reference for the first time in this issue remain in the main part of the \Kourovka Notebook", among the unsolved problems of the corresponding section. (An inquisitive reader may notice that some numbers of the problems appear neither in the main part nor in the Archive; these are the few problems that were removed at the request of the authors either as ill-conceived, or as no longer topical, for example, due to CFSG.) The abbreviation CFSG stands for The Classification of the Finite Simple Groups, which means that every finite simple non-abelian group is isomorphic either to an alternating group, or to a group of Lie type over a finite field, or to one of the twenty- six sporadic groups (see D. Gorenstein, Finite Simple Groups: The Introduction to Their Classification, Plenum Press, New York, 1982). A note \mod CFSG" in a comment means that the solution uses the CFSG. Wherever possible, references to papers published in Russian are given to their English translations. The numbering of the problems is the same as in the Russian original: sections correspond to issues, and within sections problems are ordered lexicographically by the Russian names of the authors (or Russian transliterations). The index of names may help the reader to find a particular problem. V. D. Mazurov, E. I. Khukhro Novosibirsk, January 2014 5 Problems from the 1st Issue (1965) 1.3. (Well-known problem). Can the group ring of a torsion-free group contain zero divisors? L. A. Bokut' 1.5. (Well-known problem). Does there exist a group whose group ring does not contain zero divisors and is not embeddable into a skew field? L. A. Bokut' 1.6. (A. I. Mal'cev). Is the group ring of a right-ordered group embeddable into a skew field? L. A. Bokut' 1.12. (W. Magnus). The problem of the isomorphism to the trivial group for all groups with n generators and n defining relations, where n > 2. M. D. Greendlinger ∗1.19. (A. I. Mal'cev). Which subgroups (subsets) are first order definable in a free group? Which subgroups are relatively elementarily definable in a free group? In particular, is the derived subgroup first order definable (relatively elementarily definable) in a free group? Yu. L. Ershov ∗All definable (and relatively elementarily definable) sets are described; in par- ticular, a subgroup is relatively elementarily definable if and only if it is cyclic (O. Kharlampovich, A. Myasnikov, Int. J. Algebra Comput., 23, no. 1 (2013), 91{ 110). 1.20. For which groups (classes of groups) is the lattice of normal subgroups first order definable in the lattice of all subgroups? Yu. L. Ershov 1.27. Describe the universal theory of free groups. M. I. Kargapolov 1.28. Describe the universal theory of a free nilpotent group. M. I. Kargapolov 1.31. Is a residually finite group with the maximum condition for subgroups almost polycyclic? M. I. Kargapolov 1.33. (A. I. Mal'cev). Describe the automorphism group of a free solvable group. M. I. Kargapolov 1.35. c) (A. I. Mal'cev, L. Fuchs). Do there exist simple pro-orderable groups? A group is said to be pro-orderable if each of its partial orderings can be extended to a linear ordering. M. I. Kargapolov 1.40. Is a group a nilgroup if it is the product of two normal nilsubgroups? By definition, a nilgroup is a group consisting of nilelements, in other words, of (not necessarily boundedly) Engel elements. Sh. S. Kemkhadze 1.46. What conditions ensure the normalizer of a relatively convex subgroup to be relatively convex? A. I. Kokorin 6 1st Issue (1965) 1.51. What conditions ensure a matrix group over a field (of complex numbers) to be orderable? A. I. Kokorin 1.54. Describe all linear orderings of a free metabelian group with a finite number of generators. A. I. Kokorin 1.55. Give an elementary classification of linearly ordered free groups with a fixed number of generators. A. I. Kokorin 1.65. Is the class of groups of abelian extensions of abelian groups closed under taking direct sums (A; B) 7! A ⊕ B? L. Ya. Kulikov 1.67. Suppose that G is a finitely presented group, F a free group whose rank is equal to the minimal number of generators of G, with a fixed homomorphism of F onto G with kernel N. Find a complete system of invariants of the factor-group of N by the commutator subgroup [F; N]. L. Ya. Kulikov 1.74. Describe all minimal topological groups, that is, non-discrete groups all of whose closed subgroups are discrete. The minimal locally compact groups can be described without much effort. At the same time, the problem is probably complicated in the general case. V. P. Platonov 1.86. Is it true that the identical relations of a polycyclic group have a finite basis? A. L. Shmel'kin 1.87. The same question for matrix groups (at least over a field of characteristic 0). A. L. Shmel'kin 7 Problems from the 2nd Issue (1967) 2.5. According to Plotkin, a group is called an NR-group if the set of its nil-elements coincides with the locally nilpotent radical, or, which is equivalent, if every inner automorphism of it is locally stable. Can an NR-group have a nil-automorphism that is not locally stable? V. G. Vilyatser 2.6. In an NR-group, the set of generalized central elements coincides with the nil- kernel. Is the converse true, that is, must a group be an NR-group if the set of generalized central elements coincides with the nil-kernel? V. G. Vilyatser 2.9. Do there exist regular associative operations on the class of groups satisfying the weakened Mal'cev condition (that is, monomorphisms of the factors of an arbitrary product can be glued together, generally speaking, into a homomorphism of the whole product), but not satisfying the analogous condition for epimorphisms of the factors? O. N. Golovin 2.22. a) An abstract group-theoretic property Σ is said to be radical (in our sense) if, in any group G, the subgroup Σ(G) generated by all normal Σ-subgroups is a Σ- subgroup itself (called the Σ-radical of G).
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