## June 2022 update for the 20th edition

An update for the current 20th edition of “Kourovka Notebook (Unsolved Problems in Group Theory)” is  posted.  A separate file contains the updates onlyAll the changes are also incorporated in the new version of the main PDF file Unsolved Problems in Group Theory. The Kourovka Notebook. No. 20.

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

This 20th edition contains 126 new problems and a number of comments on about 1000 problems from the previous issues. The section “Archive of Solved Problems” includes all the solved problems from the previous issues that have already been commented on in previous issues, while new solutions are found among unsolved problems in the corresponding sections.

## Kourovka Forum: talk by Dan Segal

On Thursday the 17-th of February, 2022, Dan Segal (All Souls College, Oxford) will give an online talk “Groups, rings, logic” at 20:00 Novosibirsk time (13:00 London time).

Abstract: In group theory, interesting statements about a group usually can’t be expressed in the language of first-order logic. It turns out, however, that some groups can actually be determined by their first-order properties, or, even more strongly, by a single first-order sentence. In the latter case the group is said to be finitely axiomatizable. I will describe some examples of this phenomenon (joint work with A. Nies and K. Tent). One family of results concerns axiomatizability of p-adic analytic pro-p groups, within the class of all profinite groups. Another main result is that for an adjoint simple Chevalley group of rank at least 2 and an integral domain R; the group G(R) is bi-interpretable with the ring R. This means in particular that first-order properties of the group G(R) correspond to first-order properties of the ring R. As many rings are known to be finitely axiomatizable we obtain the corresponding result for many groups; this holds in particular for every finitely generated group of the form G(R).

Chair of the session: Evgeny Khukhro

The link to the Zoom meeting: https://us02web.zoom.us/j/3167567840?pwd=aExXZmUxVmE5blYxYTVRMHhrakxuQT09

(ID 316 756 7840, Password gD11t2).

See the program updates at http://mca.nsu.ru/kourovkaforum/.

## The new 20-th edition of Kourovka Notebook

The latest 20-th edition of Kourovka Notebook (Unsolved Problems in Group Theory), Novosibirsk, 2022, has just appeared:

Unsolved Problems in Group Theory. The Kourovka Notebook. No. 20. (See a newer version in a later post.)

This 20-th issue contains 126 new problems and a number of comments on  problems from the previous issues. The section “Archive of Solved Problems” includes all the solved problems from the previous issues that have already been commented on in previous issues, while new solutions are found among unsolved problems in the corresponding sections.

## Kourovka Forum: talk by Alex Lubotzky

On Thursday the 3rd of February, 2022, Alex Lubotzky (Weizmann Institute and Hebrew University, Israel) will give an online talk “Stability and testability of permutations’ equations” at 16:00 Novosibirsk time (9:00 London time).

Abstract: Let A and B be two permutations in Sym(n) which “almost commute” — are they a small deformation of permutations that truly commute? More generally, if R is a system of word-equations in variables X=(x_1,….,x_d) and A=(A_1,…., A_d) permutations which are nearly solution; are they near true solutions? It turns out that the answer to this question depends only on the group presented by the generators X and relations R. This leads to the notions of “stable groups” and “testable groups”. We will present a few results and methods which were developed in recent years to check whether a group is stable\testable. We will also describe the connection of this subject with property testing in computer science, with the long-standing problem of whether every group is sofic and with IRS’s ( =invariant random subgroups).
A number of open questions will be presented.

Chair of the session: Mikhail Belolipetsky

The link to the Zoom meeting:  https://us02web.zoom.us/j/3167567840?pwd=aExXZmUxVmE5blYxYTVRMHhrakxuQT09 (ID 316 756 7840, Password gD11t2).

See the program updates at http://mca.nsu.ru/kourovkaforum/.

## Re-launch of Kourovka Forum

A new series of online seminars “Kourovka Forum” will be launched in  February of 2022: http://mca.nsu.ru/kourovkaforum/

The first seminar in the new series will be held on Thursday, February 3, at 16:00 Novosibirsk time (9:00 London time) via ZOOM. The first talk is by Alex Lubotzky, Stability and testability of permutations’ equations. The zoom link is https://us02web.zoom.us/j/5578968779 .

As an additional feature of the Kourovka Notebook, this series of seminars is intended to highlight the most promising research directions in group theory and, more generally, algebra, along with other relevant fields of mathematics. Clear formulations of the problems may help young researchers to focus their efforts and give them a higher chance of achieving outstanding mathematical results.

The seminars will be held once in two weeks on Thursdays. The starting time of the seminar will vary depending on the preferences of the speaker, taking into account the geography of the audience.

## October 2021 update for the 19th edition and call for new problems

An update for the current 19th edition of “Kourovka Notebook (Unsolved Problems in Group Theory)” is  posted.  A separate file contains the updates only. All the changes are also incorporated in the new version of the main PDF file Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19.

Recall that we are now preparing the new 20th edition. 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 could be “measured” and seen, the preference is usually given to concrete questions that admit “yes” or “no” answers. New problems can be sent to any of the editors by e-mail: Evgeny Khukhro khukhro@yahoo.co.uk or Victor Mazurov mazurov@math.nsc.ru . Later in the year we shall contact all the authors with “galley proofs” to make sure that the questions remain open and accurately stated (this also applies to several new problems that have already been received from various authors over the last years).

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

This 19th issue contains 111 new problems and a number of comments on about 1000 problems from the previous issues. The section “Archive of Solved Problems” includes all the solved problems from the previous issues that have already been commented on in previous issues, while new solutions are found among unsolved problems in the corresponding sections.

## Call for new problems for the new edition

Dear colleagues,

We begin the preparation of the new 20th edition of the “Kourovka Notebook”, a collection of unsolved problems in Group Theory and related areas.

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 could be “measured” and seen, the preference is usually given to concrete questions that admit “yes” or “no” answers.

New problems 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 .

Later in the year we shall contact all the authors with “galley proofs” to make sure that the questions remain open and accurately stated (this also applies to several new problems that have already been received from various authors over the last years).

Evgeny Khukhro and Victor Mazurov, Editors

## June 2021 update for the 19th edition

An update for the current 19th edition of “Kourovka Notebook (Unsolved Problems in Group Theory)” is  posted.
A separate file contains the updates only. SEE NEWER FILES IN OCTOBER UPDATE
All the changes are also incorporated in the new version of the main PDF file

### Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19. SEE NEWER FILES IN OCTOBER UPDATE

We thank everybody who help us keeping the Kourovka Notebook up to date.
Evgeny Khukhro   khukhro@yahoo.co.uk  and
Victor Mazurov  mazurov@math.nsc.ru
Editors
************************************

This 19th issue contains 111 new problems and a number of comments on about 1000 problems from the previous issues. The section “Archive of Solved Problems” includes all the solved problems from the previous issues that have already been commented on in previous issues, while new solutions are found among unsolved problems in the corresponding sections.

## December 2020 update for the 19th edition

An update for the current 19th edition of “Kourovka Notebook (Unsolved Problems in Group Theory)” is  posted.
A separate file contains the updates only.
All the changes are also incorporated in the new version of the main PDF file

### Unsolved Problems in Group Theory. The Kourovka Notebook. No. 19.

We thank everybody who help us keeping the Kourovka Notebook up to date.
Evgeny Khukhro   khukhro@yahoo.co.uk  and
Victor Mazurov  mazurov@math.nsc.ru
Editors
************************************

This 19th issue contains 111 new problems and a number of comments on about 1000 problems from the previous issues. The section “Archive of Solved Problems” includes all the solved problems from the previous issues that have already been commented on in previous issues, while new solutions are found among unsolved problems in the corresponding sections.

## Kourovka Forum further talks

The online seminars “Kourovka Forum” continue their work; see http://mca.nsu.ru/kourovkaforum/

The next seminar will be held on Thursday, 10 December, at 20:00 Novosibirsk time (=13:00 London time) via ZOOM. Efim Zelmanov (University of California, San Diego) will speak on Growth and complexity functions for basic algebraic structures. The zoom link is https://us02web.zoom.us/j/5578968779

Previous talks, with slides and recorded lectures, can be found at http://mca.nsu.ru/kourovkaforum/scheduleandabstracts/

Recall that this series of seminars is intended to highlight the most promising research directions in group theory and, more generally, algebra, along with other relevant fields of mathematics. Clear formulations of the problems may help young researchers to focus their efforts and give them a higher chance of achieving outstanding mathematical results.

The seminars are held once in two weeks on Thursdays. The starting time of the seminar will vary depending on the preferences of the speaker, taking into account the geography of the audience.