Università Cattolica del Sacro Cuore

Algebra

(M. Chiara Tamburini - 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

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)

Ongoing projects

  • PRIN06 - Teoria dei gruppi e Applicazioni

Main pubblications

  • A.Pasotti, M.A.Pellegrini,
    A new result on the problem of Buratti, Horak and Rosa, Discrete Mathematics 319 (2014), 1--14.
  • L.Di Martino, M.A.Pellegrini and A. Zalesski,
    On generators and representations of the sporadic simple groups, Comm. Algebra 42 (2014), 880--908.
  • A. Maroti and M.C.Tamburini,
    A solution to a problem of Wiegold, Communications in Algebra, Vol.41, Issue 1, (2013), pp. 34--49.
  • M.A.Pellegrini,
    2-Coverings for exceptional and sporadic simple groups, Archiv der Mathematik 101 (2013), 201--206.
  • M. Pellegrini, M.C.Tamburini and M. Vsemirnov,
    Uniform (2,k)- generation of the 4-dimensional classical groups, J. Algebra 369 (2012), 322-350.
  • M.Pellegrini, P.Shumyatsky,
    Coprime commutators in PSL(2,q), Archiv der Mathematik 99 (2012), 501--507.
  • C.Franchi,
    On minimal degrees of permutation representations of abelian quotients of finite groups, Bull. Austr. Math. J. 84 (2011), 408--413.
  • A. Maroti and M.C.Tamburini,
    Bounds for the probability of generating the symmetric and alternating groups, Arch. Math. Vol. 96 (2), 115--121 (2011).
  • C. Franchi, M. Mainardis, R. Solomon,
    A characterization of HN: addendum, J. Group Th. 13 (2010), - 151-153.
  • C.Tamburini and M.Vsemirnov,
    Irreducible (2,3,7)-subgroups of PGL(n,F), n<8, II, J.Algebra 321, no.8, 2119-2138 (2009).
  • C.Franchi, M.Mainardis, R.Solomon,
    A characterization of HN, J. Group Th. 11 (2008), 357--369.
  • C.Franchi, M.S.Lucido and M.Mainardis,
    A characterization of the sporadic group of Lyons, J.Algebra 319 (2008), 847--867.