# Algebra Seminar

The Algebra Seminar usually takes place on Wednesdays at 15:00 during term time in Nuffield G13.

## Spring 2020

### Title to be confirmed

• Daniel Kaplan, University of Birmingham
• Thursday 12 March 2020, 16:00
• Lecture Theatre B, Watson Building

Abstract not available

### Invariable generation of finite classical groups

• Eilidh McKemmie, University of Southern California
• Thursday 05 March 2020, 16:00
• Lecture Theatre B, Watson Building

We say a group is invariably generated by a subset if it forms a generating set even if an adversary is allowed to replace any elements with their conjugates. Eberhard, Ford and Green built upon the work of many others and showed that, as $n \rightarrow \infty$, the probability that $S_n$ is invariably generated by a random set of elements is bounded away from zero if there are four random elements, but goes to zero if we pick three random elements. This result gives rise to a nice Monte Carlo algorithm for computing Galois groups of polynomials. We will extend this result for $S_n$ to the finite classical groups using the correspondence between classes of maximal tori of classical groups and conjugacy classes of their Weyl groups.

### The general Sakuma Theorem

• Sergey Shpectorov, University of Birmingham
• Thursday 27 February 2020, 16:00
• Lecture Theatre B, Watson Building

The original Sakuma Theorem classifies vertex operator algebras (VOAs) generated by two Ising vectors. The properties it relies on were turned by Ivanov into the axioms of a new class of non-associative algebras called Majorana algebras and thus axial algebras were born.

The first axial version of the Sakuma theorem (for Majorana algebras) was published by Ivanov, Pasechnik, Seress and Shpectorov in 2010 and it was followed in 2015 by a more general version due to Hall, Rehren and Shpectorov, where many Majorana-specific assumptions were removed. The same year, Rehren attempted an even more general version, completely parting with Majorana restrictions and allowing arbitrary parameters a and b in the fusion rules to substitute the Majorana-specific values a=1/4 and b=1/32. He did not manage to obtain a complete classification, but he did show that the dimension of a 2-generated algebra is bounded by eight except when a=2b or a=4b.

In a joint project with Franchi and Mainardis, we reprove and generalise Rehren’s theorem to cover also the exceptional cases. In the case a=2b, we obtain the same bound, 8, on the dimension of a 2-generated algebra, although for a different spanning set. Even more interesting is the other exceptional case, where a=4b. Here we also have the upper bound eight, except when a=2 and b=1/2. In this final case, we found an unexpected example of an infinite dimensional 2-generated algebra.

### Diagonal structures and primitive permutation groups

• Cheryl Praeger, University of Western Australia
• Thursday 20 February 2020, 16:00
• Lecture Theatre B, Watson Building

Many maximal subgroups of finite symmetric groups arise as stabilisers of some structure on the point set: for example the maximal intransitive permutation groups are subset stabilisers. The primitive groups of diagonal type for a long time have seemed exceptional in this respect. Csaba Schneider and I have introduced diagonal structures which, for the first time, give a combinatorial interpretation to these primitive groups of simple diagonal type. In further work together also with Peter Cameron and Rosemary Bailey, we’ve exhibited these groups as automorphism groups of `diagonal graphs’.

### Groups of card shuffles

• Luke Morgan, University of Primorska
• Thursday 13 February 2020, 16:00
• Lecture Theatre B, Watson Building

There are two standard ways to shuffle a deck of cards, the in and out shuffles. For the in shuffle, divide the deck into two piles, hold one pile in each hand and then perfectly interlace the piles, with the top card from the left hand pile being on top of the resulting stack of cards. For the out shuffle, the top card from the right hand pile ends up on top of the resulting stack.

Standard card tricks are based on knowing what permutations of the deck of cards may be achieved just by performing the in and out shuffles. Mathematicians answer this question by solving the problem of what permutation group is generated by these two shuffles. Diaconis, Graham and Kantor were the first to solve this problem in full generality – for decks of size 2n. The answer is usually “as big as possible”, but with some rather beautiful and surprising exceptions. In this talk, I’ll explain how the number of permutations is limited, and give some hints about how to obtain different permutations of the deck. I’ll also present a more general question about a “many handed dealer” who shuffles kn cards divided into k piles.

### Constructing the automorphism group of a finite group

• Eamonn O'Brien, University of Auckland
• Thursday 06 February 2020, 16:00
• Lecture Theatre B, Watson Building

Constructing the automorphism group of a finite group remains challenging. The critical hard case is that of a finite p-group P where much effort has been invested over the past 20 years in developing recursive algorithms which work down a central series for P. If we can locate characteristic structure in P, then we can often readily solve the problem. The real challenge remains class 2 p-groups of exponent p. In this lecture we will outline algorithmic approaches and report on recent joint work which offer new hope of progress on this intractable problem.

### Isomorphic subgroups of finite solvable groups

• George Glauberman, University of Chicago
• Tuesday 21 January 2020, 16:00
• Arts Lecture Room 3

Note this seminar is on Tuesday rather than Thursday

In 2014, Moshe Newman asked the following question: If two subgroups of a finite solvable group are isomorphic and one is a maximal proper subgroup of G, must the other also be a maximal proper subgroup of G? This question is still open. I plan to discuss recent results with Geoffrey Robinson that give some sufficient conditions for an affirmative answer.

### Bases for primitive permutation groups

• Melissa Lee, Imperial College London
• Thursday 16 January 2020, 16:00
• Lecture Theatre B, Watson Building

Let $G \leq \mathrm{Sym}(\Omega)$ be a primitive permutation group. A base for $G$ is a subset $B \subseteq \Omega$ such that the pointwise stabiliser $G_B=1$. In this talk, after outlining the history and uses of bases, I will describe some recent work towards two prominent problems in the area – namely the solution to Pyber’s conjecture and the classification of primitive groups with base size two.