Group Theory for A-Level Mathematics | A-Level 数学：群论入门 考点精讲

📚 Group Theory for A-Level Mathematics | A-Level 数学：群论入门 考点精讲

Group theory is a fundamental branch of abstract algebra that explores symmetry and structure. In the A-Level Mathematics curriculum, students are introduced to the basic axioms, examples, and key theorems of group theory, providing a solid foundation for further study in pure mathematics, physics, and computer science. This article revisits the core concepts and typical exam points, complete with clear bilingual explanations and examples.

群论是抽象代数中一个基础分支，研究对称性与结构。在 A-Level 数学课程中，学生将接触群论的基本公理、例子和关键定理，为进一步学习纯数学、物理学和计算机科学打下坚实基础。本文重温核心概念与典型考点，并配以清晰的双语解释和例题。

1. What is a Group? | 什么是群？

A group is an algebraic structure consisting of a set G together with a binary operation that combines any two elements to form a third element, satisfying four fundamental properties. The operation can be addition, multiplication, composition, or any other defined rule, as long as the axioms hold. Groups are denoted by (G, ∘) where ∘ is the operation.

群是一种代数结构，由一个集合 G 和一个二元运算构成，该运算将任意两个元素组合成第三个元素，且必须满足四条基本性质。运算可以是加法、乘法、复合或其他规定好的规则，只要满足公理即可。群记作 (G, ∘)，其中 ∘ 为运算。

In A-Level exams, you will often see groups defined by operations on number sets, modular arithmetic, or symmetries of geometric shapes. Understanding the definition is the first step towards solving problems involving group properties.

在 A-Level 考试中，经常会见到用数集上的运算、模运算或几何图形的对称性来定义的群。理解定义是解决涉及群性质问题的第一步。

2. The Four Group Axioms | 群的四条公理

A set G with operation ∘ forms a group if and only if the following four axioms are satisfied:

集合 G 连同运算 ∘ 构成群，当且仅当满足以下四条公理：

Closure: For all a, b ∈ G, the result of a ∘ b is also in G.

封闭性：对任意 a, b ∈ G，a ∘ b 的结果仍在 G 内。

Associativity: For all a, b, c ∈ G, (a ∘ b) ∘ c = a ∘ (b ∘ c).

结合律：对任意 a, b, c ∈ G，(a ∘ b) ∘ c = a ∘ (b ∘ c)。

Identity element: There exists an element e ∈ G such that for every a ∈ G, e ∘ a = a ∘ e = a.

单位元：存在元素 e ∈ G，使得对每个 a ∈ G，e ∘ a = a ∘ e = a。

Inverse element: For each a ∈ G, there exists an element a⁻¹ ∈ G such that a ∘ a⁻¹ = a⁻¹ ∘ a = e.

逆元：对每个 a ∈ G，都存在 a⁻¹ ∈ G，满足 a ∘ a⁻¹ = a⁻¹ ∘ a = e。

These four axioms are the essence of a group. Checking them carefully is a common exam task, especially when the operation is given in an abstract way or via a Cayley table.

这四条公理是群的核心。仔细检查这些公理是常见的考题，尤其当运算以抽象形式或通过凯莱表给出时。

3. Examples of Groups | 群的例子

Many familiar number systems form groups under usual addition: (ℤ, +), (ℝ, +), (ℂ, +) are all groups. The identity is 0 and the inverse of a is –a. However, (ℕ, +) is not a group because there is no identity and no inverses for most elements.

许多熟悉的数系在通常加法下构成群：(ℤ, +)、(ℝ, +)、(ℂ, +) 都是群。单位元是 0，a 的逆元是 –a。但 (ℕ, +) 不是群，因为没有单位元且大多数元素没有逆元。

Under multiplication, we have to be careful. (ℝ, ×) is not a group because 0 has no multiplicative inverse. However, (ℝ \ {0}, ×) is a group. Similarly, the set of non-zero rational numbers ℚ* under multiplication forms a group.

在乘法下，要特别小心。(ℝ, ×) 不是群，因为 0 没有乘法逆元。但 (ℝ \ {0}, ×) 是群。类似地，非零有理数集 ℚ* 在乘法下构成群。

Modular arithmetic provides finite groups: (ℤₙ, +) is a group of order n. For example, ℤ₄ = {0, 1, 2, 3} under addition modulo 4 is a group. The group of units modulo n, denoted U(n) or ℤₙ*, under multiplication modulo n, is also a group for n ≥ 1.

模运算给出有限群：(ℤₙ, +) 是一个 n 阶群。例如，ℤ₄ = {0, 1, 2, 3} 在模 4 加法下是一个群。模 n 的单位群，记作 U(n) 或 ℤₙ*，在模 n 乘法下也是一个群（对 n ≥ 1）。

Symmetry groups of regular polygons (dihedral groups Dₙ) and permutation groups (Sₙ) are also studied. These illustrate non-commutative groups that often appear in A-Level questions.

正多边形的对称群（二面体群 Dₙ）和置换群（Sₙ）也会涉及。这些非交换群常在 A-Level 题目中出现。

4. Abelian Groups | 阿贝尔群

A group (G, ∘) is called abelian (or commutative) if for all a, b ∈ G, a ∘ b = b ∘ a. Many common groups are abelian, such as (ℤ, +) and (ℝ \ {0}, ×). However, matrix multiplication groups and most symmetry groups are non-abelian.

若对于所有 a, b ∈ G 有 a ∘ b = b ∘ a，则群 (G, ∘) 称为阿贝尔群（或交换群）。许多常见群都是阿贝尔群，如 (ℤ, +) 和 (ℝ \ {0}, ×)。但矩阵乘法群和大多数对称群是非阿贝尔群。

To prove a group is abelian, you must show the commutative property holds for all pairs of elements. To show it is not abelian, a single counterexample suffices.

要证明一个群是阿贝尔群，必须证明交换律对所有元素对成立。要说明它不是阿贝尔群，只需举出一个反例即可。

Exam questions often ask for a proof that a given group is abelian, or ask students to identify abelian groups from a list or a Cayley table.

考题经常要求学生证明给定群是阿贝尔群，或从列表或凯莱表中识别出阿贝尔群。

5. Cayley Tables | 凯莱表

A Cayley table (or group table) is a square table that defines the group operation for a finite group. The rows and columns are labelled by the elements of G, and the cell (i, j) contains the product of row element i and column element j.

凯莱表（或称群表）是定义有限群运算的方表。行和列用 G 的元素标记，单元格 (i, j) 内是行元素 i 与列元素 j 的运算结果。

For example, the Cayley table for (ℤ₄, +) is:

例如，(ℤ₄, +) 的凯莱表如下：

+	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

From the Cayley table, you can check group axioms: closure (all entries are in G), identity (a row identical to the header), inverses (each row contains the identity), and associativity is often implicit or must be verified separately. Commutativity is seen by symmetry about the main diagonal.

通过凯莱表可以检查群公理：封闭性（所有表项都在 G 中）、单位元（某行与表头相同）、逆元（每行都含有单位元），而结合律通常隐含或需单独验证。交换律则通过表关于主对角线对称来判断。

Questions may ask you to complete a Cayley table given some group elements and partial information, or to use the table to identify the identity and inverses.

题目可能要求根据给定的群元素和部分信息补全凯莱表，或利用表找出单位元和逆元。

6. Subgroups | 子群

A subset H of a group G is a subgroup if H is itself a group under the same operation as G. To verify a subset is a subgroup, you can use the subgroup test: H is non-empty, and for all a, b ∈ H, a ∘ b⁻¹ ∈ H.

如果群 G 的子集 H 在 G 的相同运算下自身构成群，则 H 是 G 的子群。验证子群可使用子群判别法：H 非空，且对任意 a, b ∈ H，有 a ∘ b⁻¹ ∈ H。

Every group has at least two trivial subgroups: the identity subgroup {e} and the whole group G. Interesting subgroups arise from symmetries or number systems. For example, in (ℤ, +), the set of even integers 2ℤ is a subgroup.

每个群至少有两个平凡子群：单位元子群 {e} 和整个群 G。有趣的子群来自对称性或数系。例如在 (ℤ, +) 中，偶数集 2ℤ 是一个子群。

Dihedral groups Dₙ contain cyclic subgroups (rotations) and reflection subgroups. Identifying subgroups and their orders is a key skill, leading to Lagrange’s Theorem.

二面体群 Dₙ 包含循环子群（旋转）和反射子群。识别子群及其阶是一项关键技能，进而引出拉格朗日定理。

7. Lagrange’s Theorem | 拉格朗日定理

Lagrange’s theorem states that for a finite group G, the order (number of elements) of any subgroup H divides the order of G. That is, |H| divides |G|.

拉格朗日定理指出，对于有限群 G，任意子群 H 的阶（元素个数）整除 G 的阶。即 |H| 整除 |G|。

This theorem is extremely useful for determining possible subgroup sizes. If |G| = 12, possible subgroup orders are 1, 2, 3, 4, 6, and 12. It does not guarantee a subgroup of each divisor exists, but provides a necessary condition.

该定理对确定可能的子群大小极为有用。若 |G| = 12，可能的子群阶为 1、2、3、4、6 和 12。它不保证每个因数阶的子群都存在，但给出了必要条件。

A-Level problems often ask you to list all possible orders of subgroups of a given finite group, or to prove that a group of prime order has no non-trivial proper subgroups.

A-Level 问题常要求列出给定有限群所有可能的子群阶，或证明素数阶群没有非平凡的真子群。

Lagrange’s theorem also helps in determining the order of an element: the order of an element a is the smallest positive integer n such that aⁿ = e, and it must divide the order of the group.

拉格朗日定理还有助于确定元素的阶：元素 a 的阶是满足 aⁿ = e 的最小正整数 n，且必须整除群的阶。

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 a power (or multiple) of g. Such an element g is called a generator. Cyclic groups are denoted by ⟨g⟩.

若存在元素 g ∈ G，使得 G 中每个元素都可写成 g 的幂次（或倍数），则称 G 为循环群。这样的 g 称为生成元。循环群记作 ⟨g⟩。

All cyclic groups are abelian. The group (ℤₙ, +) is cyclic with generator 1. The group of n-th roots of unity under multiplication is also cyclic. The concept of cyclic groups often appears in questions about group orders and isomorphism.

所有循环群都是阿贝尔群。(ℤₙ, +) 是循环群，生成元为 1。n 次单位根在乘法下构成的群也是循环群。循环群的概念常出现在关于群阶和同构的题目中。

To show a group is cyclic, find a generator and demonstrate that repeated application of the operation produces all elements. For instance, in ℤ₅*, the element 2 is a generator: 2¹ = 2, 2² = 4, 2³ = 3, 2⁴ = 1.

要证明一个群是循环群，需找到一个生成元并说明重复运算能产生所有元素。例如在 ℤ₅* 中，元素 2 是生成元：2¹ = 2，2² = 4，2³ = 3，2⁴ = 1。

9. Group Isomorphisms | 群同构

Two groups (G, ∘) and (H, ∗) are isomorphic if there exists a bijective function f: G → H that preserves the group operation, i.e., f(a ∘ b) = f(a) ∗ f(b) for all a, b ∈ G. Isomorphism means the groups have the same structure.

若存在双射 f: G → H 保持群运算，即对所有 a, b ∈ G 有 f(a ∘ b) = f(a) ∗ f(b)，则两个群 (G, ∘) 和 (H, ∗) 同构。同构意味着两个群具有相同的结构。

Proving isomorphism often involves constructing the mapping and checking it is bijective and operation-preserving. A standard result is that any cyclic group of order n is isomorphic to ℤₙ.

证明同构通常需要构造映射并检查其双射且保运算。一个标准结论是任意 n 阶循环群都同构于 ℤₙ。

Exam questions may ask you to show two given finite groups are isomorphic by comparing their Cayley tables, or to prove that certain properties (like being abelian) are preserved under isomorphism.

考题可能要求通过比较凯莱表来说明两个给定有限群同构，或证明某些性质（如交换性）在同构下保持不变。

10. Direct Products of Groups | 群的直积

Given two groups (G, ∘) and (H, ∗), their direct product G × H is the set of ordered pairs (g, h) with componentwise operation: (g₁, h₁) · (g₂, h₂) = (g₁ ∘ g₂, h₁ ∗ h₂). This construction yields a new group whose order is |G| × |H|.

给定两个群 (G, ∘) 和 (H, ∗)，它们的直积 G × H 是有序对 (g, h) 的集合，运算逐分量进行：(g₁, h₁) · (g₂, h₂) = (g₁ ∘ g₂, h₁ ∗ h₂)。这种构造产生一个新群，其阶为 |G| × |H|。

Direct products allow us to build larger groups from smaller ones. For example, ℤ₂ × ℤ₂ is a group of order 4 which is not cyclic; it is isomorphic to the Klein four-group. Understanding direct products is useful for classifying finite abelian groups.

直积允许我们从较小的群构建更大的群。例如 ℤ₂ × ℤ₂ 是一个 4 阶非循环群，同构于克莱因四元群。理解直积有助于对有限阿贝尔群进行分类。

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