On finite Sylow tower and σ-tower groups

Abstract

In this paper, G is a finite group and σ a partition of the set of all primes ℙ, that is, σ = {σ_i| i ∈ I}, where ℙ = ⋃_i∈I σ_i and σ_i ∩ σ_j = ∅ for all i = j; σ (G) = {σ_i| σ_i ∩ π(G) = ∅}.

We say that G is a σ-tower group if either G = 1 or G has a normal series 1 = G₀ < G₁ < · · · < G_t−1 < G_t= G such that G_k/G_k−1 is a σ_i-group, σ_i ∈ σ(G), and G/G_k and G_k−1 are σ^′_i-groups for all k = 1, . . . , t.

Now we associate with each group G = 1 a directed graph Γ = Γ_{H σ} (G) as follows: Let σ(G) = {σ₁ , σ₂ , . . . , σ_t}. Γ_{H σ} (G) has t vertices σ₁ , σ₂ , . . . , σ_t and (σ_i , σ_j) is an arc of Γ (directed from σ_i to σ_j) if and only if . And we call such a graph the Hawkes σ-graph of G.

In this paper, we study various properties of finite Sylow tower and σ-tower groups. In particular, we prove that G is a σ-tower group if and only if the Hawkes σ-graph of G has no circuits, and we describe the structure of a σ-tower group G by appealing to some other properties of the graph Γ_Hσ (G).