ON THE σ-NILPOTENT NORM AND THE σ-NILPOTENT LENGTH OF A FINITE GROUP

HU, BIN ; HUANG, JIANHONG  ; SKIBA, ALEXANDER N.

Glasgow mathematical journal, Tome 63 (2021) no. 1, pp. 121-132

Résumé

Let G be a finite group and σ = {σi| i ∈ I} some partition of the set of all primes $\Bbb{P}$. Then G is said to be: σ-primary if G is a σi-group for some i; σ-nilpotent if G = G1× ... × Gt for some σ-primary groups G1, ... , Gt; 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 lσ (G)) of G is the length of the shortest normal chain of G with σ-nilpotent factors. Let Nσ (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 Nσ (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 lσ (G/ Nσ (G)) ≤ r.

Détail

DOI : 10.1017/S0017089520000051