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Groups in Galway 2018

May 18-19, 2018

Groups in Galway has been running on an annual basis since 1978. The scope of the conference covers all areas of group theory, applications, and related fields. All who are interested are invited to attend.


POSTSCRIPT: Conference photo (with thanks to Qays Shakir)


All talks will be in AC201 on the concourse of the Arts/Science Building (number 14 on this map).

Move the cursor over a title for the talk's abstract. Click a speaker's name for pdf of their talk if available.

Friday 18 May

09.45‒10.00 Opening remarks by Professor Martin L. Newell

10.00‒10.45 John Sheekey Semifields and subspaces of matrices over finite fields Finite semifields are nonassociative division algebras over a finite field. They have been studied since the early 1900s, with Dickson constructing the first nontrivial examples. The theory was further developed by Albert and Knuth in the 1960s. Semifields correspond to various objects in both linear algebra and finite geometry. They correspond for example to a certain class of projective plane; certain types of spreads in a projective space $\mathrm{PG}(2n-1,q)$; nonsingular tensors in $\mathbb{F}_q^n\otimes \mathbb{F}_q^n\otimes \mathbb{F}_q^n$; and $n$-dimensional subspaces of $n\times n$ matrices in which every nonzero element is invertible. The last of these is of particular recent interest, as this is a special case of a maximum rank distance (MRD) code, a hot topic due to potential applications in network coding. We will provide an overview for these connections, the known constructions and classifications, and present some open problems in the area.

10.45‒11.30 Coffee/tea

11.30‒12.15 Eoin Long Recent results on set intersections The study of set intersections forms a popular topic of research in extremal combinatorics, and has a wide range of applications to different areas of mathematics. Roughly speaking, here one aims to understand how large a collection of sets can be subject to a restriction on the pairwise intersections of its elements. In this talk I will discuss some recent progress on problems of this type. Joint work with Peter Keevash.

12.30‒13.15 Maura Paterson Reciprocally-weighted external difference families Let $G$ be a finite abelian group of order $n$. An $(n,k,\lambda)$ $m$-External Difference Family (EDF) is a collection of $m$ disjoint subsets of $G$ each of size $k$, with the property that each nonzero group element occurs precisely $\lambda$ times as a difference between group elements in two different subsets from the collection. A reciprocally-weight EDF (RWEDF) is a generalisation of an EDF in which the subsets may have different sizes, and the differences are counted with a weighting given by the reciprocal of the set sizes. These were proposed in connection with the study of weak algebraic manipulation detection codes. In this talk I will motivate the definition of RWEDFs before discussing recent results including the construction of infinite families of RWEDFs.

13.15‒14.30 Lunch

14.30‒15.15 Daniel Horsley Symmetric coverings and the Bruck-Ryser-Chowla theorem A $(v,k,\lambda)$-block design is a collection of $k$-element subsets, called blocks, of a $v$-set of points such that each pair of points appears together in exactly $\lambda$ blocks. A block design is symmetric if it has exactly as many blocks as points. The Bruck-Ryser-Chowla theorem famously establishes the nonexistence of various symmetric block designs, including projective planes. In this talk I will discuss generalisations of the Bruck-Ryser-Chowla theorem to the setting of coverings where, instead of demanding that each pair of points appears together in exactly some fixed number of blocks, we only require that each pair of points appear together in at least some fixed number of blocks. The proofs of these generalisations exploit the structure of $XX^T$, where $X$ is the incidence matrix of a putative covering. In some cases it is sufficient to examine the determinants of these matrices. In others, arguments concerning rational congruence of matrices and making use of Hasse-Minkowski invariants are required.

15.30‒16.15 Bettina Eick The groups of order $p^n q$ The talk shows how the number $f_n(p,q)$ of groups of order $p^n q$ for different primes $p$ and $q$ can be determined (for fixed $n$ as a function in $p$ and $q$). Explicit functions for $f_n$ for $n \leq 5$ are exhibited.

16.15‒17.00 Coffee/tea

17.00‒17.45 Götz Pfeiffer Bisets and the double Burnside algebra of a finite group The double Burnside group $B(G, H)$ of two finite groups $G, H$ is the Grothendieck group of the category of finite $(G, H)$-bisets. Certain bisets encode relationships between the representation theories of $G$ and $H$. Bouc's biset category provides a framework for studying such relationships, it has finite groups as objects, and $B(G, H)$ as morphisms between $G$ and $H$, with composition induced by the tensor product of bisets. The endomorphism ring $B(G, G)$ is called the double Burnside ring of $G$. In contrast to the (ordinary) Burnside ring $B(G)$, the double Burnside ring $B(G, G)$ of a nontrivial group $G$ is not commutative. In general, little more is known about the structure of $B(G, G)$. In the talk I'll describe a relatively small faithful matrix representation of the rational double Burnside algebra $\mathbb{Q}B(G, G)$ for certain finite groups $G$, based on a recent decomposition of the table of marks of the direct product $G \times G$, exhibiting the cellular structure of the algebra $\mathbb{Q}B(G, G)$. This is joint work with Sejong Park.

19.00 Conference Dinner: Il Vicolo Restaurant

Saturday 19 May

10.00‒10.45 Ronan Egan Using groups to construct combinatorial structures and codes In this talk I will discuss methods of constructing combinatorial structures such as block designs from finite groups and using orbit matrices. I will also describe recent and ongoing work where generalized versions of orbit matrices are constructed from Hadamard matrices and weighing matrices using their permutation automorphism groups. This leads to the construction of self-orthogonal, self-dual, optimal and near optimal codes over finite fields.

10.45‒11.15 Coffee/tea

11.15‒12.00 Rob Craigen Signed groups, cocycles, and orthogonal matrices Independently$-$indeed literally poles apart geographically$-$during the late 1980s, two apparently very different algebraic ideas, signed groups (R. Craigen) and cocyclic index functions (W. de Launey), were introduced into the study of Hadamard (and other orthogonal) matrices and designs. But while at the conceptual level they are quite distinct, the mathematical machinery underneath both turns out to be largely the same in essence. This seems to indicate an idea whose time has come; even more so if N. Ito's independent introduction, only slightly later, of Hadamard groups, invoking this same machinery, is taken into account. Further, a substantial body of work has resulted from both starting points. Not, as one might suppose, parallel workings of the same theory but at least two mostly disjoint continua of results, different both in theoretical impact and in character, apparently illustrating how different formulations of the same machinery may lead to essentially different ends. The impact of cocyclic development of designs, expertly worked out by de Launey, Flannery, Horadam and others and may now qualify as 'well known'; Hadamard groups somewhat less so; and even less, signed groups. My goal in this talk is to familiarize my audience primarily with the latter.

12.15‒13.00 Ann Trenk The distinguishing chromatic number and NG-graphs Albertson and Collins introduced the distinguishing number of a graph as the minimum number of colors needed to color the vertices so that the only automorphism of the graph which preserves colors is the identity. The distinguishing chromatic number, $\chi_D(G)$, is defined similarly, except that the coloring must also be proper, that is, adjacent vertices must get different colors. In this talk we discuss results about $\chi_D(G)$ including characterization theorems and connections to the Nordhaus-Gaddum inequality for graphs.


There is no registration fee. However, if you intend to participate in the conference, please send an email to the organizers, stating your full name, affiliation, and whether you will attend the Dinner.


Galway can be reached by public transportation from Dublin, Shannon, and Knock airports. There are direct buses from Dublin airport to Galway city operated by Citylink and GoBus. You can also take a train from Dublin city. Bus Eireann runs buses from Dublin, Shannon, and Knock airports to Galway. Directions to NUI Galway by road can be found here.

Galway is a small city. It takes about 15 minutes to walk from the Coach Station or Railway Station to the university (Google maps direction).

NUI Galway has a number of pay-and-display parking places for visitors. Cars parked in other spaces on the NUI Galway campus and not displaying a valid parking permit will be clamped.


The following hotels and guest houses are convenient for the NUI Galway campus:

See also Discover Ireland.

The organizers are Kevin Jennings, Tobias Rossmann, and Dane Flannery.

Groups in Galway 2018 is supported by