Group of Seven/Eight (G7/G8) | 七阶与八阶群

📚 Group of Seven/Eight (G7/G8) | 七阶与八阶群

In algebra, a ‘group’ is not a political alliance but a precise mathematical structure. This article explores groups of order 7 and 8, often abbreviated as G7 and G8, through complex roots of unity, cyclic groups, and the classification of all groups of order 8.

在代数中，“群”不是政治联盟，而是一种精确的数学结构。本文通过复数单位根、循环群以及八阶群的分类，探讨七阶和八阶群，通常简写为 G7 和 G8。

Edexcel Further Mathematics students meet roots of unity when solving equations such as zⁿ = 1; these same objects form the first examples of finite groups. Understanding their structure gives a powerful link between complex numbers and abstract algebra.

Edexcel 进阶数学的学生在解 zⁿ = 1 这类方程时会遇到单位根；这些对象恰好构成有限群的第一个例子。理解它们的结构能在复数与抽象代数之间建立强有力的联系。

1. What Is a Group? | 什么是群？

A group is a set G together with a binary operation ∗ that satisfies four axioms. The operation could be addition, multiplication, composition of rotations, or any rule that combines two elements of G to produce another element of G.

群是一个集合 G 连同一种二元运算 ∗，满足四条公理。该运算可以是加法、乘法、旋转的复合，或者任何把 G 中两个元素结合成 G 中另一个元素的

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