Norberto Gavioli

Ideally constrained Lie algebras

Abstract In this paper we deal with graded Lie algebras L such that there exists a positive integer r such that for every positive integer i and for every homogeneous ideal I ⊈ L i the inclusion I ⊇ L i + r −1 holds. The solvable case and the r =1 case receive a special attention.

On the number of conjugacy classes of normalisers in a finite p -group

In 1996 Poland and Rhemtulla proved that the number ν(G) of conjugacy classesof non-normal subgroups of a non-Hamiltonian nilpotent group G is at least c − 1,where c is the nilpotency class of G. In this paper we consider the map that associatesto every conjugacy class of subgroups of a finite p-group the conjugacy class of thenormaliser of any of its representatives. In spite of the fact that this map need notbe injective, we prove that, for p odd, the number of conjugacy classes of normalisersin a finite p-group is at least c (taking into account the normaliser of the normalsubgroups). In the case of p-groups of maximal class we can find a better lowerbound that depends also on the prime p.

Metabelian thin Beauville p-groups

Abstract A non-cyclic finite p-group G is said to be thin if every normal subgroup of G lies between two consecutive terms of the lower central series and | γ i ( G ) : γ i + 1 ( G ) | ≤ p 2 {|\gamma_{i}(G):\gamma_{i+1}(G)|\leq p^{2}} for all i ≥ 1 {i\geq 1} . In this paper, we determine Beauville structures in metabelian thin p-groups.

Soluble normally constrained pro-p-groups

Abstract A pro-p-group G is said to be normally constrained (or, equivalently, of obliquity zero) if every open normal subgroup of G is trapped between two consecutive terms of the lower central series of G. In this paper infinite soluble normally constrained pro-p-groups, for an odd prime p, are shown to be 2-generated. A classification of such groups, up to the isomorphism type of their associated Lie algebra, is provided in the finite coclass case, for p > 3. Moreover, we give an example of an infinite soluble normally constrained pro-p-group whose lattice of open normal subgroups is isomorphic to that of the Nottingham group. Some general results on the structure of soluble just infinite pro-p-groups are proved on the way.