Martino Garonzi

Groups equal to a product of three conjugate subgroups

Let G be a finite non-solvable group. We prove that there exists a proper subgroup A of G such that G is the product of three conjugates of A, thus replacing an earlier upper bound of 36 with the smallest possible value. The proof relies on an equivalent formulation in terms of double cosets, and uses the following theorem which is of independent interest and wider scope: Any group G with a BN-pair and a finite Weyl group W satisfies

Covering certain wreath products with proper subgroups

G = {\left( {B{n_0}B} \right)^2} = B{B^{{n_0}}}B

Direct products of finite groups as unions of proper subgroups

where n0 is any preimage of the longest element of W. The proof of the last theorem is formulated in the dioid consisting of all unions of double cosets of B in G. Other results on minimal length product covers of a group by conjugates of a proper subgroup are given.

Factorizations of finite groups by conjugate subgroups which are solvable or nilpotent

Given a finite non-cyclic group G, call σ(G) the least number of proper subgroups of G needed to cover G. In this article, we give lower and upper bounds for σ(G) for G a group with a unique minimal normal subgroup N isomorphic to where n ≥ 5 and G/N is cyclic. We also show that σ(A 5≀C 2) = 57.

Irredundant and Minimal Covers of Finite Groups

Abstract For a non-cyclic finite group X let σ(X) be the least number of proper subgroups of X whose union is X. Precise formulas or estimates are given for σ(S ≀ Cm ) for certain non-abelian finite simple groups S where Cm is a cyclic group of order m.

Minimal length factorizations of finite simple groups of Lie type by unipotent Sylow subgroups

ABSTRACT We prove several results detecting cyclicity or nilpotency of a finite group G in terms of inequalities involving the orders of the elements of G and the orders of the elements of the cyclic group of order |G|. We prove that, among the groups of the same order, the number of cyclic subgroups is minimal for the cyclic group, and the product of the orders of the elements is maximal for the cyclic group.

On the number of conjugacy classes of a permutation group

We determine all the ways in which a direct product of two finite groups can be expressed as the set-theoretical union of proper subgroups in a family of minimal cardinality.

Finite groups with six or seven automorphism orbits

We consider factorizations of a finite group G into conjugate subgroups, G = Ax1⋯Axk for A ≤ G and x1,…,xk ∈ G, where A is nilpotent or solvable. We derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group which, for a large class of groups, is linear in the non-solvable length of G. We also show that every solvable group G is a product of at most 1 + clog |G : C| conjugates of a Carter subgroup C of G, where c is a positive real constant. Finally, using these results we obtain an upper bound on the minimal length of a nilpotent conjugate factorization of a general finite group.