Crown-based powers. In this section, we outline an approach to study the question of finding the minimal number of elements required to generate a finite group, which is due to F. Dalla Volta and X. Xxxxxxxx. So let G be a finite group, with d(G) = d > 2, and let M be a normal subgroup of G, maximal with the property that d(G/M ) = d. Then G/M needs more generators than any proper quotient of G/M , and hence, as we shall see below, G/M has a very restrictive structure. We remark that G/M is sometimes referred to in the literature as a crown for G. We describe this structure as follows: let L be a finite group, with a unique minimal normal subgroup N . If N is abelian, then assume further that N is complemented in L. Now, for a positive integer k, set Lk to be the subgroup of the direct product Lk defined as follows Lk := {(x1, x2, . . . , xk) : xi ∈ L, Nxi = Nxj for all i, j} Equivalently, Lk := diag(Lk)Nk, where diag (Lk) denotes the diagonal subgroup of Lk. The group Lk is called the crown-based power of L of size k. Note that Soc(Lk) = Nk. We can now state the theorem of Xxxxx Xxxxx and Xxxxxxxx. Theorem 3.3.1 ([14], Theorem 1.4). Let G be a finite group, with d(G) ≥ 3, which requires more generators than any of its proper quotients. Then there exists a finite group L, with a unique minimal normal subgroup N, which is either nonabelian or complemented in L, and a positive integer k ≥ 2, such that G ∼= Lk.
