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.

中文翻译：

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关于有限群的 σ-幂律范数和 σ-幂律长度

让G是一个有限群并且σ= {σ一世|一世∈一世} 所有素数集合的一些分区$\Bbb{P}$. 然后G据说是：σ-初级如果G是一个σ一世- 为一些人分组一世;σ-幂零如果G=G1× … ×G吨对于一些σ- 初级组G1, … ,G吨;σ-可溶的如果每个主要因素G是σ-基本的。我们用$G^{{\mathfrak{N}}_{\sigma}}$来表示σ-幂零残差的G，即所有正规子群的交集ñ的G和σ- 幂零商G/ñ. 如果G是σ-可溶，则σ-幂零长度（表示为lσ(G）） 的G是最短正常链的长度G和σ- 幂零因子。让ñσ(G) 是规范化器的交集σ-所有子组的幂零残差G， 那是，$${N_\sigma }(G) = \bigcap\limits_{H \le G} {{N_G}} ({H^{{_\sigma }}}).$$那么子群ñσ(G) 被称为σ-幂零范数的G. 我们研究的关系σ- 幂零长度与σ- 幂零范数G. 特别地，我们证明了σ-a 的幂零长度σ-可溶性基团G最多是r(r> 1) 当且仅当lσ(G/ñσ(G)) ≤r.