# Algebra Seminar

### Seminars are held on Thursdays at 14:00 in Room B3.02 - Mathematics Institute

#### Organisers: Inna Capdeboscq and Dmitriy Rumynin

##### Term 1 - 2018/19

**October 11: **Robert Gilman (Stevens Institute of Technology)

Title: Algorithmic complexity in finitely presented groups

Abstract: Most decision problems for finitely presented groups are

undecidable, but practical partial algorithms do exist. Generic case

complexity is often used to estimate the efficacy of these algorithms. We

will review generic case complexity, point out some shortcomings and

suggest an improvement.

**October 18: ** Geoffrey Robinson

Title: Large centralizrs in finite groups.

Abstract: We discuss recent joint work with Bob Guralnick, partly inspired by the famous Brauer-Fowler paper. We prove a number of results ( some using the classification of finite simple groups (CFSG), some not) which bound the index of the Fitting subgroup of a finite group G in terms of centralizers of certain elements of G, especially involutions. We also provide a CFSG-free bound on the maximum dimension of the fixed-point space of an involution in a finite linear group.

**January 24: **Ben Fairbairn (Birkbeck College)

Title: Invertible Generating Graphs

Abstract: Let G be a group. The generating graph of G is defined as follows: the vertices are the non-trivial elements of G with two vertices being adjoined by an edge if the corresponding pair of elements generate the group. This much-studied object is known to encode a number of generational properties of the group. In this talk we will discuss a variant recently introduced by the speaker.

**February 7: **Tim Burness (University of Bristol)

Title: The length and depth of a group

Abstract: The length of a finite group G is defined to be the maximal length of an unrefinable chain of subgroups from G to 1; this notion has been the subject of numerous papers dating back to the 1960s, especially in the context of simple groups. In recent joint work with Martin Liebeck and Aner Shalev, we study a related concept, which we call the depth of G. This is the minimal length of an unrefinable chain of subgroups from G to 1 and it is interesting to compare these two parameters. In this talk, I will focus on the depth of simple groups. In particular, I will discuss our classification of the simple groups of minimal depth, and I will explain the somewhat surprising fact that alternating groups have bounded depth. I will conclude by highlighting more general results for finite groups and I will briefly mention some recent work on analogous notions of length and depth for algebraic groups and Lie groups.

**February 14: **Laura Ciobanu Radomirovic (Heriot-Watt University)

Title: Solving equations in hyperbolic groups

Abstract: For a group G, solving equations where the coefficients are elements in G and the solutions take values in G can be seen as akin to solving Diophantine equations in number theory, answering questions from linear algebra or more generally, algebraic geometry. Moreover, the question of satisfiability of equations fits naturally into the framework of the first order theory of G.

In this talk I will discuss equations in infinite discrete groups, with emphasis on free and hyperbolic groups. In the case of hyperbolic groups I will outline the approaches of Rips & Sela, and Dahmani & Guirardel to solving equations. I will then show how the solution sets can be described in terms of formal languages, and that the latest techniques involving string compression produce optimal space complexity algorithms. This is joint work with Murray Elder.

**February 21: **Peter Symonds (University of Manchester)

Title: Cohomology and coclass

Abstract: The theory of Leedham-Green and many others describes the structure of p-groups by first dividing them up according to their coclass, where a group of order p^n and nilpotency degree c has coclass n-c. Groups of a given coclass have many structural similarities. Jon Carlson conjectured in 2005 that among the p-groups of given coclass, there should be only finitely many mod p cohomology rings up to isomorphism. He gave a proof for p=2. We prove the conjecture for all primes.

**February 28:** Robert Chamberlain (University of Warwick)

Title: Minimal Permutation Representations of Finite Groups

Abstract:

Permutation representations are commonly used to represent finite groups on a computer. The degree of such representations can significantly effect the speed of computation involving the group. As such it is often useful to put some work into finding a faithful permutation representation of least degree. This talk provides an introduction to this area of research, including many of the basic tools used to study minimal permutation representations as well as some practical results.

**March 7: **Peter Kropholler (University of South Hampton)

Title: TBA