Solvable Permutation Groups
Ákos Seress · Cambridge University Press eBooks · 2003
Strong Generators in Solvable Groups Exploiting special properties of solvable groups, [Sims, 1990] describes a method for constructing a strong generating set. Recall that a finite group G is solvable if and only if it is polycyclic , that is, there exists a sequence of elements ( y 1 , …, y r ) such that G = 〈 y 1 , …, y r 〉 and for all i ∊ [1, r – 1], y i normalizes The main idea is that given an SGS for a group H ≤ Sym(Ω) and y ∊ Sym(Ω) such that y normalizes H , an SGS for 〈 H , y 〉 can be constructed without sifting of Schreier generators. The method is based on the following observation. Lemma 7.1.1. Suppose that G = 〈 H , y 〉 ≤ Sym(Ω) and y normalizes H. For a fixed Then m is an integer and there exists h ∊ H such that z := y m h fixes α z normalizes H α ; and G α , z 〉. Proof . (i) By Lemma 6.1.7, Δ is the disjoint union of G -images of Γ. Moreover, the G -images of Γ are cyclically permuted by y and m is the smallest integer such that Γ ym = Γ. In particular, α ym Ω Γ, so there exists h Ω H with the desired property.