ON THEσ-NILPOTENT NORM AND THEσ-NILPOTENT LENGTH OF A FINITE GROUP

Abstract

Let G be a finite group and σ = {σ i | i ∈ I} some partition of the set of all primes $\Bbb{P}$ . Then G is said to be: σ-primary if G is a σ i -group for some i; σ-nilpotent if G = G1× … × G t for some σ-primary groups G1, … , G t ; σ-soluble if every chief factor of G is σ-primary. We use $G^{{\mathfrak{N}}_{\sigma}}$ to denote the σ-nilpotent residual of G, that is, the intersection of all normal subgroups N of G with σ-nilpotent quotient G/N. If G is σ-soluble, then the σ-nilpotent length (denoted by l σ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let N σ (G) be the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G, that is, $${N_\sigma }(G) = \bigcap\limits_{H \le G} {{N_G}} ({H^{{_\sigma }}}).$$ Then the subgroup N σ (G) is called the σ-nilpotent norm of G. We study the relationship of the σ-nilpotent length with the σ-nilpotent norm of G. In particular, we prove that the σ-nilpotent length of a σ-soluble group G is at most r (r > 1) if and only if l σ (G/ N σ (G)) ≤ r.

Generalized norms of groups: retrospective review and current status

“…In 2020 B. Hu, J. Huang and A. Skiba [56,134] considered the σnilpotent norm N σ (G) of a group G, which is the intersection of the normalizers of the σ-nilpotent residuals of all subgroups of G.…”