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## Research

Francesca Dalla Volta is interested in Group Theory; her research deals mainly with the following subjects:#### Permutation groups.

In [1], [2] the action of classical finite groups over finite fields (linear and projective groups over the finite field with 2 elements, orthogonal groups ) on geometrical objects is considered. In particular regular orbits are explicitly constructed for these groups.

[1], [2] are the main part of the PhD thesis of F. Dalla Volta and were partially written during a period spent in the University of East Anglia (Norwich) in 1986; the subject was suggested by J. Siemons, who with J. Key had studied the existence of regular orbits for the linear and projective groups over finite fields with more than 2 elements.#### Complete mappings and admissible groups.

This subject arises from a combinatorial problem:

*If G is a group, a bijective map f:G -->G is "complete" if the map g so defined: g(x)=xf(x), is bijective. If G admits a complete map, G is said to be admissible. It is well known (see, e.g. J. Dènes, A.D. Keedwell, Latin squares: New developments in the theory and applications, Academic Press (1985)), that G is admissible if and only if the multiplication table of G, which is a Latin square, admits an orthogonal companion.*

It is immediately verified that the identity map is a complete map for a finite group of odd order.

In 1955 Hall and Paige proved the 2-Sylow subgroups of a finite admissible group are not cyclic, and they conjectured that also the viceversa is true (that is, if the Sylow 2-subgroups of G are not cyclic, G is admissible).

F. Dalla Volta and N. Gavioli in [9] and [12] studied the problem of existence of complete mappings for linear and projective groups. In [16], following an approach by Aschbacher, they found how is a possible minimal counter-example to the Hall and Paige conjecture.#### Generators, eulerian functions, presentation rank, probability of generating finite groups.

Denote by i(G) the minimal number of involutions generating a finite group G and by d(G) the minimal number of elements generating G. It is well known that, if G is not dihedral, i(G)>2. In [1],[4],[5],[6], the problem to determine i(G), when G belongs to some family of finite classical groups is studied. This problem is related to that of determining d(G); really, it is easy to recognise that if a finite simple group is generated by 3 involutions, then it may be generated by 2 elements (and it is known that any finite simple group is 2-generated).

In [10], with A. Lucchini, it is proved that if G is an almost simple group, that is, it is an automorphisms group of a finite simple group S, then d(G) =max {2, d(G/S)}. This has as consequence that an almost simple group has presentation rank=0. In a successive paper [13], the m-minimally generated groups are determined: a group is said to be m-minimally generated, if d(G)=m, but d(G/N)<m for each proper quotient of G. As a consequence, the groups which are minimal respect to the property of having rank presentation >0 are studied.

To prove the main results in [13], the eulerian functions G(n) are considered; this leads to consider the probability that n elements randomly chosen generate a group G; this has suggested to compare the probability of generating a simple group S by n elements with the probability of generating an automorphisms group of S by n elements.#### Minimally irreducible groups.

In [18], [19] the authors consider minimally irreducible groups of degree a prime, or the product of 2 primes. This subject is one of the possible problems to deal with, in the more general context of the study of linear groups (see, e.g. V. Landazuri, G. Seitz, "On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974) 418-443, and P.H. Tiep, A.E. Zalesskii, Minimal Characters of the finite classical groups", Comm. in Algebra, 24(6) 2093-2167).