Maximal Irredundant Coverings Of Some Finite Groups

The aim of this research is to contribute further results on the coverings of some finite groups. Only non-cyclic groups are considered in the study of group coverings. Since no group can be covered by only two of its proper subgroups, a covering should consist of at least 3 of its proper subgroups. If a covering contains n (proper) subgroups, then the set of these subgroups is called an n-covering. The covering of a group G is called minimal if it consists of the least number of proper subgroups among all coverings for the group; i.e. if the minimal covering consists of m proper subgroups then the notation used is s(G) = m. A covering of a group is called irredundant if no proper subset of the covering also covers the group. Obviously, every minimal covering is irredundant but the converse is not true in general. If the members of the covering are all maximal normal subgroups of a group G, then the covering is called a maximal covering. Let D be the intersection of all members in the covering. Then the covering is said to have core-free intersection if the core of D is the trivial subgroup. A maximal irredundant n-covering with core-free intersection is known as a Cn-covering and a group with this type of covering is known as a Cn-group. This study focuses only on the minimal covering of the symmetric group S9 and the dihedral group Dn for odd n � 3; on the characterization of p-groups having a Cn-covering for n 2 f10;11;12g; and the characterization of nilpotent groups having a Cn-covering for n 2 f9;10;11;12g. In this thesis, a lower bound and an upper bound for s(S9) is established

L2-invariants of nonuniform lattices in semisimple Lie groups

We compute L2-invariants of certain nonuniform lattices in semisimple Lie groups by means of the Borel-Serre compactification of arithmetically defined locally symmetric spaces. The main results give new estimates for Novikov-Shubin numbers and vanishing L2-torsion for lattices in groups with even deficiency. We discuss applications to Gromov's Zero-in-the-Spectrum Conjecture as well as to a proportionality conjecture for the L2-torsion of measure equivalent groups.Comment: 35 pages, 2 figure

Covering monolithic groups with proper subgroups

Given a finite non-cyclic group $G$, call $\sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\sigma(G) < \sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.Comment: I wrote this paper for the Proceedings of the conference "Ischia Group Theory 2012" (March, 26th - 29th 2012