On finite Sylow tower and σ-tower groups

In this paper, G is a finite group and σ a partition of the set of all primes ℙ, that is, σ = {σi | i ∈ I}, where ℙ = ∪i∈I σi and σi ∩ σj = Ø for all i ≠ j; σ(G) = {σi | σi ∩ π(G) ≠ Ø}.We say that G is a σ-tower group if either G = 1 or G has a normal series 1 = G0 < G1 < · · · < Gt−1 < Gt = G such that Gk/Gk−1 is a σi-group, σi ∈ σ(G), and G/Gk and Gk−1 are σ′i-groups for all k = 1, . . . , t.Now we associate with each group G ≠ 1 a directed graph Γ = ΓHσ(G) as follows: Let σ(G) = {σ1, σ2, . . . , σt}. ΓHσ(G) has t vertices σ1, σ2, . . . , σt and (σi, σj) is an arc of Γ (directed from σi to σj) 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 ΓHσ(G).