当前位置: X-MOL 学术Arch. Math. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
On nilpotency of higher commutator subgroups of a finite soluble group
Archiv der Mathematik ( IF 0.5 ) Pub Date : 2020-08-19 , DOI: 10.1007/s00013-020-01514-8
Josean da Silva Alves , Pavel Shumyatsky

Let G be a finite soluble group and $$G^{(k)}$$ G ( k ) the k th term of the derived series of G . We prove that $$G^{(k)}$$ G ( k ) is nilpotent if and only if $$|ab|=|a||b|$$ | a b | = | a | | b | for any $$\delta _k$$ δ k -values $$a,b\in G$$ a , b ∈ G of coprime orders. In the course of the proof, we establish the following result of independent interest: let P be a Sylow p -subgroup of G . Then $$P\cap G^{(k)}$$ P ∩ G ( k ) is generated by $$\delta _k$$ δ k -values contained in P (Lemma 2.5 ). This is related to the so-called focal subgroup theorem.

中文翻译:

关于有限可溶群的高交换子子群的幂零性

设 G 是一个有限可溶群,$$G^{(k)}$$G ( k ) 是 G 的导出级数的第 k 项。我们证明 $$G^{(k)}$$ G ( k ) 是幂零当且仅当 $$|ab|=|a||b|$$ | AB | = | | | 乙 | 对于任何 $$\delta _k$$ δ k -values $$a,b\in G$$ a , b ∈ G 的互质阶数。在证明过程中,我们建立以下独立兴趣的结果:设 P 是 G 的 Sylow p -子群。然后 $$P\cap G^{(k)}$$ P ∩ G ( k ) 由包含在 P 中的 $$\delta _k$$ δ k 值生成(引理 2.5 )。这与所谓的焦点子群定理有关。
更新日期:2020-08-19
down
wechat
bug