Some notes on the rank of a finite soluble group

Let G be a finite group and let σ = {σ _i|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A ₁ × ⋯ × A _r , where A _i is a ${\sigma _{{i_j}}}$-group for some i _j = i _j (A _i ). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σ _i -subgroup of G for some i ∈ I and ℋ has exactly one Hall σ _i -subgroup of G for every i such that σ _i ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AH ^x = H ^x A for all H ∈ ℋ and x ∈ G. The symbol r(G) (r _p (G)) denotes the rank (p-rank) of G.

Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and r _p (G) ≤ n + r _p (H) − 1 for all H ∈ ℋ and odd p ∈ π(H).