On finite 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 } \sigma=\{\sigma_{i}\mid i\in I\} , where P = ⋃ i ∈ I σ i \mathbb{P}=\bigcup_{i\in I}\sigma_{i} and σ i ∩ σ j = ∅ \sigma_{i}\cap\sigma_{j}=\emptyset for all i ≠ j i\neq j . If 𝑛 is an integer, we write σ ⁢ ( n ) = { σ i ∣ σ i ∩ π ⁢ ( n ) ≠ ∅ } \sigma(n)=\{\sigma_{i}\mid\sigma_{i}\cap\pi(n)\neq\emptyset\} and σ ⁢ ( G ) = σ ⁢ ( | G | ) \sigma(G)=\sigma(\lvert G\rvert) . A group 𝐺 is said to be 𝜎-primary if 𝐺 is a σ i \sigma_{i} -group for some i = i ⁢ ( G ) i=i(G) and 𝜎-soluble if every chief factor of 𝐺 is 𝜎-primary. We say that 𝐺 is a 𝜎-tower group if either G = 1 G=1 or 𝐺 has a normal series 1 = G 0 < G 1 < ⋯ < G t - 1 < G t = G 1=G_{0}