Abstract: (Joint work with Pierre-Emmanuel Caprace and Timothée Marquis.) Compactly generated locally compact groups G have a well-behaved notion of ends, generalizing the number of ends of a finitely generated group: G has 0,1,2 or infinitely many ends, and having more than one end is associated to a certain kind of action on a tree (not necessarily of finite degree). It can also happen that the action on the tree is micro-supported, meaning that for each half-tree, there is an element fixing that half-tree pointwise but acting nontrivially on the opposite half-tree. The existence of micro-supported actions is in turn closely related to the structure of locally normal subgroups (closed subgroups with open normalizer) and has further implications for global properties of G, for instance it often leads to a nonamenable action of G on the Cantor space.
We find a sufficient condition in terms of a conjugacy class of compact subgroups for G to act on a tree in a way that shows G has infinitely many ends, and the action is also micro-supported. As an application, we obtain a connection between local and large-scale structure, for a class of groups acting on buildings that are obtained from Kac–Moody groups over finite fields. This is a class of totally disconnected locally compact groups where much is known about the group on a large scale (via the geometry of the building), but the structure of the profinite open subgroups is still mysterious.
]]>Abstract: Let be a rational expression of degree three over the finite field . We count the irreducible polynomials in , of a given degree, which have the form for some . As an example of application of our results, we recover the number of irreducible transformation shift registers of order three, which were computed by Jiang and Yang in 2017.
]]>Evgeny Khukhro and Sandro Mattarei are two of the main speakers at the forthcoming international conference “Topics in Algebra (a conference in honor of Andrea Caranti and Carlo Maria Scoppola)”, which will take place on 1–2 September 2022 in Trento (Italy).
New paper by Evgeny Khukhro (Univ. of Lincoln) and Wolfgang Moens† (University of Vienna) “Fitting height of finite groups admitting a fixed-point-free automorphism satisfying an additional polynomial identity” has been accepted for publication in Journal of Algebra.
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†The second author died in May 2022.
Abstract: Let be a non-zero polynomial with integer coefficients. An automorphism of a group is said to satisfy the elementary abelian identity if the linear transformation induced by on every characteristic elementary abelian section of is annihilated by . We prove that if a finite (soluble) group admits a fixed-point-free automorphism satisfying an elementary abelian identity , where is a primitive polynomial, then the Fitting height of is bounded in terms of . We also prove that if is any non-zero polynomial and is a -group for a finite set of primes depending only on , then the Fitting height of is bounded in terms of the number irr of different irreducible factors in the decomposition of . These bounds for the Fitting height are stronger than the well-known bounds in terms of the composition length of when or irr is small in comparison with .
]]>This Thursday 30th of June the School of Mathematics & Physics of Lincoln is offering an event for year 12 students studying level 3 mathematics who are looking to find out more about studying maths at university. Among the offered activities, there will be some problem solving as well, a talk about admissions procedures for studying maths at university, and a masterclass. Check the detailed programme below.
10:30 – 10:45 | Welcome and H&S (Dr Matthew Booth) |
10:45 – 11:15 | Applying to university (Dr Martin Greenall) |
11:15 – 11:45 | Subject talk: “Group Theory: Algebra of Transformations” (Prof. Evgeny Khukhro) |
11:45 – 12:00 | An undergraduate student’s perspective (Leah Ward) |
12:00 – 12:30 | Lunch break |
12:30 – 13:45 | Problem solving session (Dr Matthew Booth) |
13:15 – 14:00 | Activity: “Estimating pi by rolling dice” (Prof Andrei Zvelindovsky & Owen Cockroft) |
14:00 – 14:30 | Q&A |
Prof. Evgeny Khukhro: Group Theory: Algebra of Transformations: Every mathematical object has its own group of structure-preserving transformations. Group Theory studies groups in their own right; it is a dynamic rich area of Algebra, with many open problems, where new results and methods appear all the time. The lecture introduces the idea of a group, illustrated by examples of groups of transformations and by calculations in an abstract group. Breakthrough results on greater commutativity of group operations are explained, along with links to recent group theory research by Lincoln algebraists. The lecture slides are available here.
Prof Andrei Zvelindovsky & Owen Cockroft: Estimating pi by rolling dice: This activity demonstrates a method for estimating pi from the results of rolling dice, that involves some advanced mathematics including complex analysis and prime numbers.
For more information, access https://amsp.org.uk/events/details/10004.
]]>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.
Gustavo A. Fernández-Alcober and Şükran Gül visited Lincoln in May 2019. Marialaura Noce visited Lincoln in September 2019. Elena Di Domenico visited Lincoln in November and December 2019, and Iker de las Heras visited Lincoln in January 2020.
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