# Workshop on Permutation Groups: Methods and Applications

January 12th - 14th, 2017
Bielefeld University
• Michael Giudici (University of Western Australia)
Semiprimitive groups
A transitive permutation group is called semiprimitive if every normal subgroup is either transitive or semiregular. This class of groups includes all primitive, quasiprimitive and innately transitive groups. They were introduced by Bereczky and Maróti and have received recent attention due to a conjecture of Potočnik, Spiga and Verret that generalises the Weiss conjecture for locally primitive graphs. In this talk I will discuss recent work with Luke Morgan that develops a general theory for the structure of semiprimitive groups.

• We explain how to generalize the well-known cycle decomposition of elements of the symmetric groups to elements of Coxeter groups satisfying a certain property which is always fulfilled in the case of the symmetric groups. More precisely, we show that an element having a reduced factorization into reflections such that the reflections in this factorization generate a parabolic subgroup admits an analogue of the cycle decomposition. We then give a characterization of these elements (joint with B. Baumeister, K. Roberts and P. Wegener) in finite Coxeter groups in terms of the Hurwitz action on their set of reduced factorizations into reflections. These elements are called (parabolic) quasi-Coxeter elements", because they are very close to being (parabolic) Coxeter elements.
• In this talk I will present the current knowledge on the modular character tables of the Fischer groups and some computational methods used to construct these character tables.
• Martin Liebeck (Imperial College London)
Orbits of linear groups
I will present various results concerning the orbit sizes of finite linear groups $G < {\rm GL}(n,F)$ ($F$ a field) on the set of vectors. I will also discuss some applications of these to representation theory and permutation group theory.
• The maximal subgroup problem is central to finite group theory and has wide ranging applications in number theory and geometry. Aschbacher O'Nan Scott showed that a solution of the general maximal subgroup problem for finite groups requires the classification of the maximal subgroups of the finite almost simple groups and the determination of the cohomology groups $H^1(G,V)$ for all quasisimple $G$ and all irreducible $\mathbb{F}_p G$-modules $V$ and all primes $p$. The primary open problem in the classification of the maximal subgroups of the finite almost simple groups is the determination of the overgroups of finite quasisimple groups which act irreducibly on the natural module of a finite classical group. We will discuss how recent advances in the representation theory of the finite simple groups can be brought to bear on the maximal subgroup problem of the finite classical groups.
• Gunter Malle (TU Kaiserslautern)
Zeros of characters
We present a result on zeros of irreducible characters of quasi-simple groups on $p$-singular elements. This relies on a Murnaghan-Nakayama type formula for values of unipotent characters of classical groups. We then sketch applications to questions of Robinson on endotrivial modules, of Pellegrini-Zalesskii on $p$-vanishing characters and of Koshitani on first Cartan invariants. This is joint work with C. Lassueur and F. Lübeck.
• Attila Maróti (Rényi Institute, Budapest)
Pyber's base size conjecture
Building on earlier papers of several authors, we establish the following statement. There exists a universal constant $c > 0$ such that the minimal base size $b(G)$ of a primitive permutation group $G$ of degree $n$ satisfies $b(G) < 45 (\log |G| / \log n) + c$. This finishes the proof of Pyber's base size conjecture. An ingredient of the proof is that for the distinguishing number $d(G)$ (in the sense of Albertson and Collins) of a transitive permutation group $G$ of degree $n > 1$ we have $d(G) \leqslant 48 \sqrt[n]{|G|}$. This is joint work with Hülya Duyan and Zoltán Halasi.
• Effective algorithms to handle permutation groups are required for a variety of different applications. For example, algorithms for answering questions about matrix groups defined over a finite field may need to decide whether a group encountered en route is isomorphic to some alternating group of unknown degree. We discuss a selection of probabilistic algorithms for permutation groups. As the reliability of such algorithms rests on a detailed analysis of the proportions of certain kinds of elements, we present some of these underlying results.
• Benjamin Nill (University of Magdeburg)
Permutation groups and polytopes
There is a long and fascinating history of connections between groups and polytopes. In this talk we will mostly focus on the following simple construction. Given a permutation group $G$ as a subgroup of the set of $n$ by $n$ permutation matrices, its associated permutation polytope $P(G)$ is the convex hull of the elements in $G$. For instance, the convex hull of all $n$ by $n$ permutation matrices is the permutation polytope of the symmetric group $S_n$, called the Birkhoff polytope, which is of significance in optimization and statistics. In this way, permutation groups give rise to polytopes with interesting properties, while conversely experimental observations about these geometric objects lead to non-trivial problems for permutation groups. In this non-technical survey on work by Baumeister, Friese, Gr\"{u}ninger, Guralnick, Haase, Ladisch, Perkinson, et al. we will ask (and partially answer) some natural questions: What can we say about the structure of permutation polytopes? When are they isomorphic and in what sense? What conclusions about a permutation group can we make from its associated polytope?
• Cheryl Praeger (University of Western Australia)
Three-transpositions, graphs, and groupoids
In 1969 Fischer published his wonderful theory of three-transposition groups, in particular constructing the three Fischer sporadic finite simple groups. In addition to its crucial role in identifying the three sporadic simple groups, the three-transposition theory caught the imagination of mathematicians working in combinatorics and finite geometry, as well as group theory. I will attempt to trace several mathematical paths where three transposition theory influenced the development, or simply appeared unexpectedly.
• The Classification of Finite Simple Groups (CFSG) is a monumental achievement and a seemingly indispensable tool in modern finite group theory. By now there are a few results which can be used to bypass this tool in a number of cases, most notably a theorem of Larsen and Pink which describes the structure of finite linear groups of bounded dimension over finite fields. In a few cases more ad hoc arguments can be used to delete the use of CFSG from the proofs of significant results. The talk will among others discuss a recent example due to the speaker: how to obtain a CFSG-free version of Babai's quasipolynomial Graph Isomorphism algorithm by proving a Weird Lemma about permutation groups.
• Colva Roney-Dougal (University of St Andrews)
Generating sets of finite groups
It is well known that generating sets for groups are far more complicated than generating sets for, say, vector spaces. The latter satisfy the exchange axiom, and hence any two irredundant sets have the same cardinality. According to the Burnside Basis Theorem, a similar property holds for groups of prime power order.
We define a new sequence of relations on the elements of a finite group, one for each positive integer $r$, where two elements of a finite group are equivalent if each can be substituted for the other in any $r$-element generating set. These relations become finer as $r$ increases: we define a new numerical group invariant to be the value of $r$ at which they stabilise. We are able to characterise this value of $r$ for all soluble groups, and give upper and lower bounds for all finite groups.
The generating graph of a $2$-generated finite group is a graph whose vertices are the group elements, and whose edges are the $2$-element generating sets. Our new relations yield a precise description of the automorphism group of the generating graph of any finite soluble group with nonzero spread.
• Aner Shalev (Hebrew University of Jerusalem)
Some conjectures and some miracles in finite simple groups
Finite simple groups have some remarkable properties, which sometimes look too good to be true. I will discuss some of these properties, and related classical conjectures of Ore, of Thompson, as well as a proof of conjectures of Gowers and Viola on mixing and complexity in simple groups.
I will also discuss a recent work with Burness and Liebeck relating certain generation questions to Number Theory and the existence of special primes.
The talk will be accessible to a wide audience.
• Katrin Tent (Münster)
Sharply 2-transitive groups
The existence of sharply 2-transitive groups without regular normal subgroup was a longstanding open problem. Recently constructions have been given, at least in certain characteristics. We will survey the current state of the art and explain some constructions and their limitations.
• We first survey old results on the subgroup structure of the classical groups ${\rm SL}(n,K)$, ${\rm Sp}(2n,K)$ and ${\rm SO}(n,K)$, in particular, a reduction theorem for classifying the maximal subgroups and the methodology developed by Dynkin and Seitz for completing this classification. We then turn to more recent offshoots of this work, exploiting the techniques and developing others, to study restrictions of irreducible representations to certain maximal subgroups.
The more recent results discussed appear in work of Burness, Cavallin, Ford, Ghandour, Liebeck, Marion and Seitz
• t.b.a.
• A well-developed branch of finite permutation group theory studies properties of certain classes of finite permutation groups $G$ as a function of their degree, $n$. For example, one can choose a class and ask: How large is $|G|$ in terms of $n$? How many generators will $G$ need in terms of $n$? If one chooses elements of $G$ at random, with replacement, then how long will one have to wait before a generating set is found? What if one asks these questions with generators" replaced by invariable generators"?
In this talk, we will address these questions in the class of transitive permutation groups.
• Robert A. Wilson
The Monster
t.b.a.