A-Level WJEC Maths: Introduction to Group Theory – Key Concepts Explained | A-Level WJEC 数学：群论入门考点精讲

📚 A-Level WJEC Maths: Introduction to Group Theory – Key Concepts Explained | A-Level WJEC 数学：群论入门考点精讲

Group theory is a central topic in abstract algebra and forms a key part of the WJEC A-Level Further Mathematics syllabus. Understanding the axioms that define a group, recognising common examples, and applying core theorems such as Lagrange’s Theorem are essential skills. This revision guide walks you through the fundamental concepts of group theory, from binary operations to isomorphisms, with clear explanations and worked illustrations to support your exam preparation.

群论是抽象代数的核心内容，也是 WJEC A-Level 进阶数学大纲的重要组成部分。理解定义群的公理，识别常见群例子，并应用拉格朗日定理等核心定理是必备技能。本复习指南将带你梳理群论的基础概念，从二元运算到同构，配合清晰的解释和示例，助你备战考试。

1. Binary Operations and the Group Axioms | 二元运算与群公理

A binary operation ∗ on a set S is a rule that takes any two elements a, b ∈ S and produces a unique element a ∗ b ∈ S. The operation is said to be closed if the result stays within S. Group theory is built on four key axioms that a set and operation must satisfy to form a group.

集合 S 上的二元运算 ∗ 是一种规则，对 S 中任意两个元素 a, b，产生唯一元素 a ∗ b ∈ S。如果运算结果始终落在 S 内部，则称该运算封闭。群论建立在一个集合和运算必须满足的四条核心公理之上。

The axioms are: closure (for all a,b ∈ S, a ∗ b ∈ S), associativity (a ∗ (b ∗ c) = (a ∗ b) ∗ c), existence of an identity element e such that e ∗ a = a ∗ e = a for all a, and existence of an inverse a⁻¹ for each a satisfying a ∗ a⁻¹ = a⁻¹ ∗ a = e.

公理包括：封闭性（对所有a,b ∈ S，a ∗ b ∈ S），结合律（a ∗ (b ∗ c) = (a ∗ b) ∗ c），存在单位元 e 使得对所有 a 有 e ∗ a = a ∗ e = a，以及每个 a 存在逆元 a⁻¹ 满足 a ∗ a⁻¹ = a⁻¹ ∗ a = e。

2. Definition of a Group | 群的定义

A group is an ordered pair (G, ∗) where G is a non‑empty set and ∗ is a binary operation on G that satisfies closure, associativity, identity and inverse properties. We usually denote the group simply by G when the operation is understood. If the set is finite, we call it a finite group; otherwise it is an infinite group.

群是一个有序对 (G, ∗)，其中 G 是非空集合，∗ 是 G 上的二元运算，满足封闭性、结合律、单位元与逆元。当运算明确时，常直接用 G 表示群。若集合有限，则称为有限群；否则为无限群。

For example, the integers under addition (ℤ, +) form a group: addition is closed, associative, 0 is the identity, and the inverse of n is −n. The natural numbers ℕ under addition do not form a group because they lack inverses (negative numbers are not in ℕ).

例如，整数在加法下 (ℤ, +) 构成群：加法封闭、可结合，0 是单位元，n 的逆元是 −n。自然数 ℕ 在加法下不构成群，因为缺少逆元（负数不在 ℕ 中）。

3. Abelian Groups | 阿贝尔群

If a group operation also satisfies commutativity, i.e. a ∗ b = b ∗ a for all a,b ∈ G, then the group is called Abelian (or commutative). Many familiar groups, such as (ℤ, +), (ℝ, +) and the set of 2×2 matrices under addition, are Abelian.

如果群运算还满足交换律，即对所有 a,b ∈ G 有 a ∗ b = b ∗ a，则这个群称为阿贝尔群（或交换群）。许多熟悉的群，如 (ℤ, +)、(ℝ, +) 以及 2×2 矩阵在加法下，都是阿贝尔群。

The symmetric group S₃ (all permutations of three objects under composition) is a classic non‑Abelian group, because swapping elements in different orders can yield different results. Recognising whether a group is Abelian is a common exam requirement.

对称群 S₃（三个对象的全排列在复合下）是一个经典的非阿贝尔群，因为以不同顺序交换元素会得到不同结果。识别一个群是否为阿贝尔群是考试中的常见要求。

4. Cayley Tables | 凯莱表

A Cayley table (or group table) displays the result of applying the binary operation to every pair of elements in a finite group. It provides a complete picture of the group structure, making it easy to check closure, identify the identity element and find inverses.

凯莱表（或群表）展示了有限群中每一对元素进行二元运算的结果。它完整呈现了群结构，便于检查封闭性、识别单位元以及找出逆元。

Below is the Cayley table for the Klein four‑group V₄ = {e, a, b, c}, where every element is its own inverse and the group is Abelian. The identity e appears exactly once in each row and column, confirming the Sudoku‑like property of group tables.

以下是克莱因四元群 V₄ = {e, a, b, c} 的凯莱表，每个元素都是自身的逆元，且该群是阿贝尔群。单位元 e 在每行每列恰好出现一次，印证了群表类似数独的性质。

∗	e	a	b	c
e	e	a	b	c
a	a	e	c	b
b	b	c	e	a
c	c	b	a	e

5. Examples of Groups: Modular Arithmetic | 群的例子：模算术群

Modular arithmetic provides an important family of finite Abelian groups. The set ℤₙ = {0,1,2,…,n−1} under addition modulo n forms the cyclic group of order n, denoted (ℤₙ, +ₙ). Closure and inverses are straightforward: the inverse of k is n−k mod n.

模算术提供了一族重要的有限阿贝尔群。集合 ℤₙ = {0,1,2,…,n−1} 在模 n 加法下构成 n 阶循环群，记作 (ℤₙ, +ₙ)。封闭性和逆元都很直接：k 的逆元是 n−k mod n。

Another example is the group of units modulo n, denoted U(n) or ℤₙ*, which consists of integers less than n that are coprime to n, under multiplication modulo n. For instance, U(5) = {1,2,3,4} with multiplication modulo 5 is a group of order 4.

另一个例子是模 n 的单位群，记作 U(n) 或 ℤₙ*，由小于 n 且与 n 互质的整数在模 n 乘法下构成。例如，U(5) = {1,2,3,4} 在模 5 乘法下是一个 4 阶群。

6. Subgroups | 子群

A subset H of a group G is a subgroup, written H ≤ G, if H is itself a group under the same operation as G. This requires H to be non‑empty, closed under the operation, and to contain the inverse of each of its elements.

群 G 的子集 H 称为子群，记作 H ≤ G，如果 H 在 G 的运算下自身也构成群。这要求 H 非空，对运算封闭，并且包含每个元素的逆元。

The subgroup test simplifies verification: a non‑empty subset H is a subgroup of G if and only if for all a,b ∈ H, the product a ∗ b⁻¹ ∈ H. Common subgroups include the trivial subgroup {e}, the whole group G, and the centre Z(G).

子群检验法简化了验证：非空子集 H 是 G 的子群当且仅当对所有 a,b ∈ H，有 a ∗ b⁻¹ ∈ H。常见的子群包括平凡子群 {e}、整个群 G 以及中心 Z(G)。

7. Cyclic Groups and Generators | 循环群与生成元

A group G is cyclic if there exists an element g ∈ G such that every element of G can be written as a power of g (using the group operation). The element g is called a generator, and we write G = ⟨g⟩. Both ℤ under addition and ℤₙ under addition modulo n are cyclic.

如果存在元素 g ∈ G，使得 G 中每个元素都能写成 g 的幂（用群运算），则称 G 为循环群。元素 g 称为生成元，记作 G = ⟨g⟩。加法下的 ℤ 和模 n 加法下的 ℤₙ 都是循环群。

For example, ℤ₆ = {0,1,2,3,4,5} is generated by 1 or by 5, since repeatedly adding 1 (mod 6) cycles through all elements. Not all groups are cyclic; the Klein four‑group V₄ has no single generator, so it is not cyclic.

例如，ℤ₆ = {0,1,2,3,4,5} 由 1 或 5 生成，因为反复加 1（模 6）会遍历所有元素。并非所有群都是循环群；克莱因四元群 V₄ 没有一个生成元，因此它不是循环群。

8. Order of a Group and Order of an Element | 群的阶与元素的阶

The order of a group G, denoted |G|, is the number of elements in the set G. For an element a ∈ G, the order ord(a) is the smallest positive integer n such that aⁿ = e, where e is the identity. If no such n exists, the element has infinite order.

群 G 的阶，记作 |G|，是集合 G 中元素的个数。对于元素 a ∈ G，其阶 ord(a) 是使得 aⁿ = e 的最小正整数 n，其中 e 是单位元。如果不存在这样的 n，则元素具有无限阶。

In a finite group, every element has finite order, and the order of an element divides the group order (a consequence of Lagrange’s Theorem). In (ℤ, +), the element 1 has infinite order because no positive integer multiple of 1 gives 0.

在有限群中，每个元素都有有限阶，且元素的阶整除群的阶（这是拉格朗日定理的推论）。在 (ℤ, +) 中，元素 1 具有无限阶，因为没有任何正整数倍的 1 得到 0。

9. Lagrange’s Theorem | 拉格朗日定理

Lagrange’s Theorem states that for a finite group G, the order of any subgroup H divides the order of G. Symbolically, |H| divides |G|. This is a powerful tool for determining possible subgroup orders and proving that certain subsets cannot be subgroups.

拉格朗日定理指出，对于有限群 G，任何子群 H 的阶整除 G 的阶。符号表示为 |H| 整除 |G|。这是确定可能子群阶数以及证明某个子集不可能成为子群的有力工具。

A direct corollary is that the order of any element a divides the order of the group, because the cyclic subgroup ⟨a⟩ generated by a has order ord(a). For instance, in a group of order 8, elements can only have orders 1, 2, 4 or 8.

一个直接推论是，任何元素 a 的阶整除群的阶，因为由 a 生成的循环子群 ⟨a⟩ 的阶为 ord(a)。例如，在一个 8 阶群里，元素的阶只能是 1、2、4 或 8。

10. Isomorphisms | 同构

Two groups (G, ∗) and (H, ◦) are isomorphic if there exists a bijective function φ: G → H that preserves the group operation, i.e. φ(a ∗ b) = φ(a) ◦ φ(b). Isomorphic groups have identical algebraic structures, even though their elements and operations may look different.

如果存在一个双射函数 φ: G → H 保持群运算，即 φ(a ∗ b) = φ(a) ◦ φ(b)，则两个群 (G, ∗) 和 (H, ◦) 同构。同构的群具有相同的代数结构，尽管它们的元素和运算看起来可能不同。

For example, the group U(5) = {1,2,3,4} under multiplication mod 5 is isomorphic to the cyclic group ℤ₄ under addition mod 4. The mapping φ(1)=0, φ(2)=1, φ(4)=2, φ(3)=3 is an isomorphism. Recognising isomorphisms helps to classify groups efficiently.

例如，在模 5 乘法下的单位群 U(5) = {1,2,3,4} 同构于模 4 加法下的循环群 ℤ₄。映射 φ(1)=0, φ(2)=1, φ(4)=2, φ(3)=3 就是一个同构映射。识别同构有助于高效地对群进行分类。

φ(ab mod 5) ≡ φ(a) + φ(b) (mod 4) — preservation of operation

φ(ab mod 5) ≡ φ(a) + φ(b) (mod 4) — 运算保持

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