📚 AS Mathematics: Introduction to Group Theory – Key Points | AS 数学：群论入门 考点精讲
Group theory is a fundamental part of abstract algebra studied in AS Further Mathematics. It provides a formal framework for analysing symmetry and structure. Mastering the axioms, key examples, Cayley tables, subgroups, and cyclic groups is essential for exam success. This article covers the core concepts you need to know.
群论是 AS 进阶数学中抽象代数的基础内容，它为分析对称性和结构提供了形式化框架。掌握公理、典型示例、凯莱表、子群以及循环群是考试成功的关键。本文涵盖你需要掌握的核心概念。
1. What is a Group? | 什么是群？
A group is a set G together with a binary operation ∗ that satisfies four specific axioms. The operation combines any two elements of G to produce another element of G, and the structure (G, ∗) is written to denote a group. Groups appear naturally in number systems, symmetries of shapes, and transformations.
群是一个集合 G 连同满足四条特定公理的二元运算 ∗。该运算将 G 中任意两个元素结合产生 G 中的另一个元素，记作 (G, ∗) 表示该群。群天然地出现在数系、图形的对称性与变换中。
In essence, a group abstracts the idea of symmetry: a set of objects that can be combined in a consistent way, with an operation that is invertible and has an identity. For example, the integers under addition form a group, but the natural numbers under addition do not.
本质上，群抽象了对称性的概念：一组可以以一致方式结合的对象，运算具备可逆性和单位元。例如，整数在加法下构成群，但自然数在加法下不构成群。
2. The Four Group Axioms | 群的四大公理
Closure: For all a, b ∈ G, the result a ∗ b must be in G. This ensures the operation never leaves the set.
封闭性： 对所有 a, b ∈ G，结果 a ∗ b 必须仍在 G 中。这保证运算不会离开集合。
Associativity: (a ∗ b) ∗ c = a ∗ (b ∗ c) for every a, b, c ∈ G. The order in which operations are performed does not change the result. Many familiar operations are associative, but Cayley table tasks often assume it.
结合律： 对每个 a, b, c ∈ G 有 (a ∗ b) ∗ c = a ∗ (b ∗ c)。运算的执行顺序不改变结果。许多熟悉的运算满足结合律，但在凯莱表题目中常假设成立。
Identity element: There exists an element e ∈ G such that e ∗ a = a ∗ e = a for all a ∈ G. The identity acts as a neutral element. In many examples the identity is 0 for addition or 1 for multiplication.
单位元： 存在一个元素 e ∈ G，使得对所有 a ∈ G 有 e ∗ a = a ∗ e = a。单位元起到中性元素的作用。许多例子中加法单位元是 0，乘法单位元是 1。
Inverse element: For each a ∈ G there exists an element a⁻¹ ∈ G such that a ∗ a⁻¹ = a⁻¹ ∗ a = e. Every element must have an inverse that brings it back to the identity.
逆元： 对每个 a ∈ G，存在元素 a⁻¹ ∈ G 满足 a ∗ a⁻¹ = a⁻¹ ∗ a = e。每个元素必须有一个把它带回单位元的逆元。
3. Examples of Groups and Non-Groups | 群与非群的例子
The set of integers ℤ under addition is a group: closure holds, addition is associative, 0 is the identity, and the inverse of k is −k.
整数集 ℤ 在加法下是群：加法封闭，满足结合律，0 是单位元，k 的逆元是 −k。
The set of non-zero real numbers ℝ* under multiplication is a group. The identity is 1, and the inverse of x is 1/x. However, all integers under multiplication is not a group because most elements lack multiplicative inverses that are integers (e.g., 2 has no integer inverse).
非零实数集 ℝ* 在乘法下是群。单位元是 1，x 的逆元是 1/x。然而，所有整数在乘法下不是群，因为大多数元素缺乏整数逆元（例如 2 没有整数逆元）。
The set of residue classes {0,1,2,3} under addition modulo 4, denoted ℤ₄, is a finite group of order 4. The Cayley table of this group will be examined later.
剩余类集合 {0,1,2,3} 在模 4 加法下记作 ℤ₄，是一个 4 阶有限群。该群的凯莱表将在后面分析。
4. Abelian Groups | 阿贝尔群
A group (G, ∗) is called abelian (or commutative) if the operation is commutative: a ∗ b = b ∗ a for all a, b ∈ G. Many common groups are abelian, including (ℤ, +), (ℝ*, ×), and cyclic groups.
如果群 (G, ∗) 的运算满足交换律，即对所有 a, b ∈ G 有 a ∗ b = b ∗ a，则称其为阿贝尔群（或交换群）。许多常见群是阿贝尔群，包括 (ℤ, +)、(ℝ*, ×) 和循环群。
Not every group is abelian. The smallest non-abelian group is the symmetric group S₃, which has 6 elements representing all permutations of three objects. For AS level you may encounter questions asking you to prove whether a given group is abelian, often by inspecting its Cayley table.
并非所有群都是阿贝尔群。最小的非阿贝尔群是对称群 S₃，它有 6 个元素，表示三个对象的所有排列。在 AS 层次你可能会遇到要求证明给定群是否阿贝尔的题目，通常通过检查其凯莱表实现。
5. Order of a Group and Order of an Element | 群的阶与元素的阶
The order of a group G, written |G|, is the number of elements in the set G. If |G| is finite, G is called a finite group. For example, |ℤ₄| = 4.
群 G 的阶，记作 |G|，是集合 G 中元素的个数。若 |G| 有限，则 G 称为有限群。例如 |ℤ₄| = 4。
The order of an element a in G is the smallest positive integer n such that aⁿ = e (where aⁿ means a ∗ a ∗ … ∗ a, n times) using the group operation. If no such n exists, the element has infinite order. In ℤ₄ under addition, the element 2 has order 2 because 2 + 2 ≡ 0 mod 4, and the identity is 0. The element 1 has order 4.
元素 a 在群中的阶是使得 aⁿ = e 的最小正整数 n（其中 aⁿ 表示 n 次群运算）。若不存在这样的 n，该元素为无限阶。在 ℤ₄ 加法下，元素 2 的阶为 2，因为 2 + 2 ≡ 0 mod 4，单位元为 0。元素 1 的阶为 4。
6. Cayley Tables | 凯莱表
A Cayley table is a grid that displays the results of the group operation for all pairs of elements. It is an essential tool for verifying closure, finding the identity, checking inverses, and testing commutativity for small finite groups.
凯莱表是一个展示所有元素对运算结果的网格。它是验证封闭性、寻找单位元、检查逆元以及检验小有限群交换性的重要工具。
Below is the Cayley table for ℤ₄ under addition modulo 4. The identity is 0, and the table is symmetric, confirming the group is abelian.
下面是 ℤ₄ 在模 4 加法下的凯莱表。单位元为 0，表格对称，确认该群是阿贝尔群。
| + | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 |
| 1 | 1 | 2 | 3 | 0 |
| 2 | 2 | 3 | 0 | 1 |
| 3 | 3 | 0 | 1 | 2 |
When you are asked to complete a Cayley table for a group, use the fact that each row and each column must contain every element exactly once (a property derived from the cancellation laws). This helps deduce missing entries.
当你被要求补全一个群的凯莱表时，利用每行每列必须恰包含每个元素一次这一事实（来自消去律的性质）。这有助于推断空缺项。
7. Subgroups | 子群
A subset H of a group G is a subgroup, denoted H ≤ G, if H is itself a group under the same operation ∗ restricted to H. This means H must satisfy all four axioms. However, we can use a simplified test: H is a subgroup if it is non-empty and for any a, b ∈ H, the product a ∗ b⁻¹ ∈ H (subgroup test).
群 G 的子集 H 若本身在相同运算 ∗ 下构成群，则称 H 为 G 的子群，记作 H ≤ G。这意味着 H 必须满足全部四条公理。但我们可以使用简化检验：H 是非空的，且对任意 a, b ∈ H，乘积 a ∗ b⁻¹ ∈ H（子群检验法）。
For finite groups, closure alone guarantees that a non-empty subset H is a subgroup. A popular exam task is to list all subgroups of a given group. For ℤ₄, the subgroups are {0}, {0,2} and the whole group ℤ₄. Notice that the orders of the proper subgroups divide the order of the group (a glimpse of Lagrange’s theorem).
对于有限群，仅封闭性就足以保证非空子集 H 为子群。常见的考试任务是列出给定群的所有子群。对于 ℤ₄，子群有 {0}、{0,2} 以及整个群 ℤ₄。注意，真子群的阶整除群的阶（这是拉格朗日定理的体现）。
8. Cyclic Groups | 循环群
A group G is cyclic if there exists an element g ∈ G such that every element of G can be written as gⁿ for some integer n. Such an element g is called a generator, and we write G = ⟨g⟩. All cyclic groups are abelian.
若存在元素 g ∈ G 使得 G 中的每个元素都可写为某整数 n 下的 gⁿ，则称群 G 为循环群。这样的元素 g 称为生成元，记作 G = ⟨g⟩。所有循环群都是阿贝尔群。
A finite cyclic group of order n is isomorphic to ℤₙ under addition modulo n, often denoted Cₙ. For example, ℤ₄ = ⟨1⟩ because adding 1 repeatedly generates all elements. The element 3 also generates ℤ₄ because 3, 3+3=2, 3+2=1, 3+1=0, and 3 has order 4.
n 阶有限循环群同构于模 n 加法的 ℤₙ，常记作 Cₙ。例如 ℤ₄ = ⟨1⟩，因为反复加 1 生成所有元素。元素 3 也能生成 ℤ₄，因为 3 的阶为 4，且 3, 3+3=2, 3+2=1, 3+1=0。
The number of generators of a cyclic group of order n is φ(n), where φ is Euler’s totient function. In ℤ₄, n=4, φ(4)=2, and indeed 1 and 3 are generators. This can appear in harder AS questions.
n 阶循环群的生成元个数为 φ(n)，其中 φ 为欧拉函数。在 ℤ₄ 中，n=4，φ(4)=2，确实 1 和 3 是生成元。这可能出现在较难的 AS 题目中。
9. Group Isomorphism | 群同构
Two groups (G, ∗) and (H, ∘) are isomorphic if there exists a bijective function f: G → H such that f(a ∗ b) = f(a) ∘ f(b) for all a, b ∈ G. Isomorphism means the groups have the same structure; their Cayley tables are identical up to relabelling of elements.
两个群 (G, ∗) 和 (H, ∘) 同构，若存在双射 f: G → H 使得对所有 a, b ∈ G 有 f(a ∗ b) = f(a) ∘ f(b)。同构意味着群具有相同的结构；它们的凯莱表在重新标记元素后完全一致。
For AS level, you may need to show two groups are isomorphic by finding a mapping that preserves the operation, or to recognise that a group of order 4 must be either cyclic (ℤ₄) or the Klein four-group V₄. Practice matching Cayley tables to identify isomorphisms.
在 AS 级别，你可能需要通过找到一个保持运算的映射来证明两个群同构，或者识别出 4 阶群要么是循环群 ℤ₄，要么是克莱因四元群 V₄。练习匹配凯莱表以辨认同构。
10. Key Properties and Common Mistakes | 重要性质及常见错误
The identity element e in a group is unique. If there were two identities e₁ and e₂, then e₁ = e₁e₂ = e₂. Similarly, each element has exactly one inverse. These proofs often
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导