全部书籍

1 Primary & Middle School Mathematics1–9 年级在线 2 High School Mathematics10–12 年级在线 3 University Mathematics — Year 1本科一年级在线 4 University Mathematics — Year 2本科二年级在线 5 University Mathematics — Year 3本科三年级在线

1 Primary & Middle School Physics1–9 年级在线 2 High School Physics10–12 年级在线 3 University Physics — Year 1本科一年级在线 4 University Physics — Year 2本科二年级在线 5 University Physics — Year 3本科三年级在线

1 School Chemistry — Grades 1 to 121–12 年级在线 2 University Chemistry — Year 1本科一年级在线 3 University Chemistry — Year 2本科二年级在线 4 University Chemistry — Year 3本科三年级在线

1 Primary & Middle School Biology1–9 年级在线 2 High School Biology10–12 年级在线 3 University Biology — Year 1本科一年级在线 4 University Biology — Year 2本科二年级在线 5 University Biology — Year 3本科三年级在线

1 Primary & Middle School Earth and Space Sciences1–9 年级在线 2 High School Earth and Space Sciences10–12 年级在线 3 University Earth and Space Sciences — Year 1本科一年级在线 4 University Earth and Space Sciences — Year 2本科二年级在线 5 University Earth and Space Sciences — Year 3本科三年级在线

1 Markets I: The Ecosystem and Exchange-Traded Markets市场在线 2 Markets II: Rates, FX and Credit市场在线 3 Markets III: Commodities, Energy and Crypto市场在线 4 Quantitative Methods方法在线 5 Derivatives and Volatility衍生品在线 6 Rates, Credit, XVA and Risk利率、信用与风险在线 7 Research Craft: Predictors, Backtests, Measurement, Portfolios研究在线 8 Strategies I: Equities and Futures策略在线 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desks策略在线 10 Microstructure and Execution交易执行在线 11 Market Making and High-Frequency Trading做市在线 12 Machine Learning for Markets机器学习在线 13 Low-Latency Software技术在线 14 Networks, Hardware and Trading Infrastructure技术在线 15 Research, Data and Risk Platforms技术在线 16 The Desk and the Firm公司运营在线 17 The Industry: Firms, Roles and Careers职业发展在线 18 Interviews I: Maths, Probability and Speed职业发展在线 19 Interviews II: Markets, Trading Games and the Final Round职业发展在线 20 Interviews III: Code职业发展在线

全部游戏 Math Fuse Word Reactor Atlas Dash

应用关于

Coach

English Français Español Nederlands Português हिन्दी العربية Bahasa Indonesia 简体中文 বাংলা

登录开始阅读

书籍暂未推出简体中文版——当前显示英文版。

Mathematics · 术语表

什么是 commutator？

又称： derived subgroup · solvable group

Definition 1.28 University Mathematics — Year 3 · 第 1 章 — Group Theory

The commutator of $x, y \in G$ is $[x,y] = xyx^{-1}y^{-1}$ ; the derived subgroup $D(G)$ is the subgroup generated by all commutators. The derived series is $D^0(G) = G$ , $D^{i+1}(G) = D(D^i(G))$ , and $G$ is solvable if $D^n(G) = \{e\}$ for some $n$ .

示例

Example 1.30

Abelian groups are solvable. $p$ -groups are solvable, by induction on the order: $Z(G) \neq \{e\}$ and $G/Z(G)$ is a smaller $p$ -group. $S_3$ and $S_4$ are solvable: $S_4 \trianglerighteq A_4 \trianglerighteq V \trianglerighteq \{e\}$ , where $V = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}$ is the Klein group of double transpositions (normal in $S_4$ : a union of conjugacy classes), with abelian quotients $\Z/2\Z$ , $\Z/3\Z$ , $V$ . In Chapter 4, “the general equation of degree $n$ is solvable by radicals” will literally mean “ $S_n$ is a solvable group”. Whence the importance of the next definition.

在原文中阅读 →