On finite 𝜎-tower groups

Abstract

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}<G_{1}<\cdots<G_{t-1}<G_{t}=G such that G i / G i - 1 G_{i}/G_{i-1} is a σ i \sigma_{i} -group, σ i ∈ σ ⁢ ( G ) \sigma_{i}\in\sigma(G) , and G / G i G/G_{i} and G i - 1 G_{i-1} are σ i ′ \sigma_{i}^{\prime} -groups for all i = 1 , … , t i=1,\ldots,t . A subgroup 𝐴 of 𝐺 is said to be 𝜎-subnormal in 𝐺 if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A t = G A=A_{0}\leq A_{1}\leq\cdots\leq A_{t}=G such that either A i - 1 ⁢ ⊴ ⁢ A i A_{i-1}\trianglelefteq A_{i} or A i / ( A i - 1 ) A i A_{i}/(A_{i-1})_{A_{i}} is 𝜎-primary for all i = 1 , … , t i=1,\ldots,t . In this paper, answering to Question 4.8 in [A. N. Skiba, On 𝜎-subnormal and 𝜎-permutable subgroups of finite groups, J. Algebra 436 (2015), 1–16], we prove that a 𝜎-soluble group G ≠ 1 G\neq 1 with | σ ⁢ ( G ) | = n \lvert\sigma(G)\rvert=n is a 𝜎-tower group if each of its ( n + 1 ) (n+1) -maximal subgroups is 𝜎-subnormal in 𝐺.