Title: A chain of normalizers in the Sylow 2-subgroup of Sym(2^n).

Abstract: The normalizer of a regular elementary abelian 2-subgroup in the symmetric group Sym(2^n) is the affine group AGL(2^n). Due to some questions arising from differential cryptanalysis of block cyphers, we started studying the normalizer chain in Sym(2^n) starting from the Sylow 2-subgroup of AGL(2^n). Computational results suggested a possible connection with the sequence (a_m) of the number of partitions of m into distinct parts already studied by Euler. We could prove indeed that for i\le n-2 the the base 2 logarithm index of the i-th term of the chain in the (i-1)-th one is equal to (i-2)-th term of the partial sum sequence arising from (a_m). This result rely on a combinatorial structure, that will be described, involving a family of commutators in a suitable set of generators of the Sylow 2-subgroup of Sym(2^n).

]]>This talk was the first of the joint Lincoln-Lund Algebra Seminar.

Here are some scenes from the talk.

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Title: The conjugacy problem for ascending HNN-extensions of free groups.

Abstract: The conjugacy problem is one of the three Fundamental Problems for groups, as posed by Dehn over 100 years ago. It is undecidable in general, but known to be decidable for many important classes of groups including free-by-cyclic groups. In this talk, I will explain my proof of the decidability of the conjugacy problem for ascending HNN-extensions of free groups. This class of groups generalises free-by-cyclic groups, and I will draw parallels, and explain the differences between the two settings.

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Recall that “Kourovka Notebook” is a famous collection of open problems in group theory proposed by hundreds of mathematicians from all over the world, published every 2-4 years since 1965.

**The preparation of the new 20th edition is now underway.** 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 e-mail: Evgeny Khukhro khukhro@yahoo.co.uk or Victor Mazurov mazurov@math.nsc.ru .

Lincoln algebraists and Anitha continue close research contacts and collaborations, in particular, establishing a joint Lincoln–Lund algebra research seminar. Anitha has also been appointed a Visiting Senior Fellow in Algebra at University of Lincoln.

]]>The algebra research seminar resumes work in 2021–22 as a joint venture of Charlotte Scott Centre for Algebra of University of Lincoln (UK) and Centre for Mathematical Sciences of Lund University (Sweden). Preliminary dates and times of some of the nearest talks are

27th Oct, 3pm UK time — Alan Logan (Heriot-Watt Univ.)

10th Nov, 3pm UK time — Charles Cox (Bristol Univ.)

24th Nov, 3pm UK time — Angela Carnevale (Galway Univ.)

The seminars will take place via Zoom, and the details will appear in individual talks announcements later.

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*Abstract*: *Nottingham algebras* are a class of just-infinite-dimensional, modular, -graded Lie algebras, which includes the graded Lie algebra associated to the Nottingham group with respect to its lower central series. Homogeneous components of a Nottingham algebra have dimension one or two, and in the latter case they are called *diamonds*. The first diamond occurs in degree 1, and the second occurs in degree *q*, a power of the characteristic. Many examples of Nottingham algebras are known, in which each diamond past the first can be assigned a *type*, either belonging to the underlying field or equal to .

A prospective classification of Nottingham algebras requires describing all possible diamond patterns. In this paper we establish some crucial contributions towards that goal. One is showing that all diamonds, past the first, of an *arbitrary* Nottingham algebra *L* can be assigned a type, in such a way that the degrees and types of the diamonds completely describe *L*. At the same time we prove that the difference in degrees of any two consecutive diamonds in any Nottingham algebra equals *q-*1. As a side-product of our investigation, we classify the Nottingham algebras where all diamonds have type .

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 .

Recall that “Kourovka Notebook” is a famous collection of open problems in group theory proposed by hundreds of mathematicians from all over the world, published every 2-4 years since 1965.

Sponsored by School of Mathematics and Physics of the University of Lincoln and Sobolev Institute of Mathematics, Novosibirsk.

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