## Algebra

(Clara Franchi - Marco Antonio Pellegrini)

Our research activity, centred in group theory, concerns the following aspects.

**Characterization of Sporadic Simple groups**

One of the most important achievements of the mathematics of the 20th century is the classification of finite simple groups. It has been done thankful to the contribution of a great number of researchers, during about more than forty years. In the classification theorem, the 26 sporadic simple groups represent something strange, since their properties are so particular that we do not have an homogeneous description of them all, yet. Moreover, they most probably have (still) mysterious connections with physics. The aim of our project is to study some of the sporadic simple groups under different viewpoints.

**Generations of grups**

A central problem in group theory, also in view of its applications, is the following: given a group G, find a set of generators for G, having certain properties. We study certain aspects of this vast problem (in particular the (2,3)-generation and the Hurwitz generation), especially for classical groups over a commutative ring and simple groups of Lie type. The methods are advanced linear algebra, the theory of group representations, the classification of maximal subgroups of finite classical groups. For the computational aspects we use the algebraic software Magma-Cayley.

**Representation theory**

The representation theory allows us to construct a finite group as a matrix group in such a way that we can actually work with its elements. Several information about the group can be obtained considering only the traces of these matrices. The study of these traces is called character theory. We are interested in the properties of characters and representations of finite simple groups or finite groups of Lie type. For instance, we are studying the eigenvalues of the matrices representing elements of simple groups and classifying particular characters of finite groups of Lie type vanishing on the set of the non-trivial unipotent elements.

### Key words

- group theory
- finite groups
- simple sporadic groups

### Contacts

- Clara Franchi (clara.franchi@unicatt.it)
- Marco Antonio Pellegrini (marcoantonio.pellegrini@unicatt.it)

### Collaborations

- Prof. Lino Di Martino (Università di Milano-Bicocca),
- Prof. A.A. Ivanov (Imperial College, London),
- Prof. Mario Mainardis (Università di Udine),
- Dr. Attila Maroti (Università di Budapest, Ungheria),
- Dott. M. Mecchia (Università di Trieste),
- Dott.ssa Anita Pasotti (Università di Brescia),
- Prof. Schpectorov (University of Birmingham),
- Prof. Ron Solomon (Ohio - State Univesity),
- Dr. E.P. Vdovin (Università di Novosibirsk),
- Prof. Maxim Vsemirnov (Steklov Institute, San Pietroburgo),
- Prof. Alexandre Zalesski (Università di Milano-Bicocca)

### Past projects

- PRIN06 - Teoria dei gruppi e Applicazioni