Evgeny Khukhro announced a start for preparation of the new 19th edition of the “Kourovka Notebook” — the famous collection of unsolved problems in Group Theory and related areas. This publication originated in Novosibirsk in 1965, since then gained world-wide popularity and now includes more than 1000 problems by about 400 authors from all over the world. Nowadays it is published on the web but still retains discrete issues appearing every 3–4 years, with more frequent updates online.

For more than 50 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! (Of course, it is often easier to propose a new problem than to solved an old one…)

Everybody is welcome to propose new problems to be included in the new edition. Problems may “belong” to those who propose them, or otherwise. In the latter case, one can indicate the author(s) of the problem (if different from the person proposing), or simply that this is a “well-known problem”. In order that the progress would be “measured” and seen, the preference is usually given to concrete questions that admit “yes” or “no” answers.

The Editors also welcome any other comments on, or/and solutions of, existing problems; the current version on Arxiv incorporates all comments so far.

The problems and comments can be sent to any of the Editors (preferably by e-mail):

Evgeny Khukhro khukhro@yahoo.co.uk or Victor Mazurov mazurov@math.nsc.ru

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Sandro Mattarei and Roberto Tauraso, *From generating series to polynomial congruences,* 24 pages, submitted, 2017;* *arXiv:1703.02322.

Abstract: Consider an ordinary generating function , of an integer sequence of some combinatorial relevance, and assume that it admits a closed form . Various instances are known where the corresponding truncated sum , with a power of a prime , also admits a closed form representation when viewed modulo . Such a representation for the truncated sum modulo frequently bears a resemblance with the shape of , despite being typically proved through independent arguments. One of the simplest examples is the congruence being a finite match for the well-known generating function .

We develop a method which allows one to directly infer the closed-form representation of the truncated sum from the closed form of the series for a significant class of series involving central binomial coefficients. In particular, we collect various known such series whose closed-form representation involves polylogarithms , and after supplementing them with some new ones we obtain closed-forms modulo for the corresponding truncated sums, in terms of finite polylogarithms .

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December 29, 2016 – 14:55
“Kourovka Notebook” is a collection of open problems in Group Theory proposed by more than 300 mathematicians from all over the world. It has been published every 2-4 years since 1965. This is the 18th edition, which contains 120 new problems and a number of comments on about 1000 problems from the previous editions.

This new update for the current 18th edition has been posted at http://arxiv.org/abs/1401.0300. In addition, for convenience of the readers, all the changes made since the 18th edition first appeared are also listed separately in 18upd-e. We thank all the people who help us keeping Kourovka Notebook up to date.

Evgeny Khukhro khukhro@yahoo.co.uk and Victor Mazurov mazurov@math.nsc.ru, Editors

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The paper by **Evgeny Khukhro and Pavel Shumyatsky ***Engel-type subgroups and length parameters of finite groups *has been accepted for publication* *in *Israel Journal of Mathematics*. The results of the paper have been obtained in collaboration between Evgeny Khukhro of University of Lincoln and Pavel Shumyatsky of University of Brasilia, with Evgeny’s visits to Brasilia supported by CNPq-Brazil grant within the Brazilian Scientific Mobility Program “Ciências sem Fronteiras”.

Abstract: Let be an element of a finite group . For a positive integer , let be the subgroup generated by all commutators over , where is repeated times. By Baer’s theorem, if , then belongs to the Fitting subgroup . We generalize this theorem in terms of certain length parameters of . For soluble we prove that if, for some , the Fitting height of is equal to , then belongs to the th Fitting subgroup . For nonsoluble the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height of a finite group is the least number such that , where , and is the inverse image of the generalized Fitting subgroup . Let be the number of prime factors of counting multiplicities. It is proved that if, for some , the generalized Fitting height of is equal to , then belongs to , where depends only on and . The nonsoluble length of a finite group is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if , then belongs to a normal subgroup whose nonsoluble length is bounded in terms of and . We also state conjectures of stronger results independent of and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.

Full text: https://arxiv.org/abs/1506.00233

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The paper by **E. I. Khukhro and P. Shumyatsky, ****Almost Engel finite and profinite groups** has been accepted for publication in the *International Journal of Algebra and Computation*, ISSN 0218-1967, see also arXiv:1512.06097.

Abstract: Let be an element of a group . For a positive integer , let be the subgroup generated by all commutators over , where is repeated times. We prove that if is a profinite group such that for every there is such that is finite, then has a finite normal subgroup such that is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group , we prove that if, for some , for all , then the order of the nilpotent residual is bounded in terms of .

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New update for the current 18th edition has been posted at http://arxiv.org/abs/1401.0300 .

In addition, for convenience of the readers, all the changes made since the 18th edition first appeared are also listed separately in 18upd-e . We thank all the people who help us keeping Kourovka Notebook up to date.

Evgeny Khukhro khukhro@yahoo.co.uk and Victor Mazurov mazurov@math.nsc.ru , Editors

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E. I. Khukhro and P. Shumyatsky, *Almost Engel finite and profinite groups, *submitted, 2015;* *arXiv:1512.06097.

Abstract: Let be an element of a group . For a positive integer , let be the subgroup generated by all commutators over , where is repeated times. We prove that if is a profinite group such that for every there is such that is finite, then has a finite normal subgroup such that is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group , we prove that if, for some , for all , then the order of the nilpotent residual is bounded in terms of .

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Evgeny Khukhro, Pavel Shumyatsky, *Engel-type subgroups and length parameters of finite groups,*

http://arxiv.org/abs/1506.00233

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New update for the current 18th edition has been posted at http://arxiv.org/abs/1401.0300 .

In addition, for convenience of the readers, all the changes made since the 18th edition first appeared are also listed separately in 18upd-e . We thank all the people who help us keeping Kourovka Notebook up to date. Evgeny Khukhro khukhro@yahoo.co.uk and Victor Mazurov mazurov@math.nsc.ru , Editors

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