📚 Edexcel Mathematics: Introduction to Group Theory Key Points Revision | Edexcel 数学：群论入门 考点精讲
In Edexcel Further Mathematics, group theory introduces a rigorous algebraic structure that underpins symmetry, number theory, and many advanced mathematical concepts. Mastering the core definitions, axioms, and fundamental theorems is essential for success in the examination. This revision guide systematically covers every key point, from binary operations to Cayley tables, with clear bilingual explanations and practical examples that align with the Edexcel specification.
在 Edexcel 进阶数学中，群论引入了一种严谨的代数结构，为对称性、数论以及许多高等数学概念奠定基础。掌握核心定义、公理和基本定理对于考试成功至关重要。本考点精讲从二元运算到凯莱表，系统梳理每一个关键点，提供清晰的中英双语解释和符合 Edexcel 考纲的实用示例。
1. Binary Operations and Closure | 二元运算与封闭性
A binary operation * on a set S is a rule that assigns to each ordered pair (a, b) ∈ S × S a unique element denoted a * b. The operation is said to be closed on S if for all a, b ∈ S, the result a * b is also an element of S. Closure is the first requirement for any group structure.
集合 S 上的二元运算 * 是指对每个有序对 (a,b)∈S×S 指定唯一元素 a*b 的规则。如果对所有 a,b∈S，结果 a*b 仍属于 S，则称该运算在 S 上封闭。封闭性是任何群结构的首要条件。
For example, standard addition + is closed on the set of integers ℤ, because the sum of any two integers is an integer. However, subtraction is not closed on the set of natural numbers ℕ = {1,2,3,…}, since 2 − 5 = −3, which is not in ℕ.
例如，普通加法在整数集 ℤ 上是封闭的，因为任意两个整数之和仍是整数。但减法在自然数集 ℕ={1,2,3,…} 上不封闭，因为 2−5=−3 不在 ℕ 中。
You will often need to check whether a given operation defined by a table or formula is closed on a finite set. If any result falls outside the set, the system cannot form a group.
考试中常需检验由表格或公式定义的运算在有限集上是否封闭。只要有一个结果落在集合外，该系统就不可能构成群。
2. Group Axioms | 群公理
A non-empty set G together with a binary operation * forms a group (G, *) if it satisfies the following four axioms:
一个非空集合 G 连同二元运算 * 构成群 (G, *)，如果满足以下四条公理：
G1 (Closure): For all a, b ∈ G, a * b ∈ G.
G1 (封闭性)：对所有 a,b∈G，a*b∈G。
G2 (Associativity): For all a, b, c ∈ G, (a * b) * c = a * (b * c).
G2 (结合律)：对所有 a,b,c∈G，(a*b)*c = a*(b*c)。
G3 (Identity): There exists an element e ∈ G such that for every a ∈ G, e * a = a * e = a.
G3 (单位元)：存在 e∈G，使得对每个 a∈G 都有 e*a = a*e = a。
G4 (Inverse): For each a ∈ G, there exists an element a⁻¹ ∈ G such that a * a⁻¹ = a⁻¹ * a = e.
G4 (逆元)：对每个 a∈G，存在 a⁻¹∈G 满足 a*a⁻¹ = a⁻¹*a = e。
These axioms must all be verified when demonstrating that a given set and operation form a group. Note that commutativity is not required; a group satisfying a * b = b * a for all elements is called Abelian (dealt with later).
证明某一集合和运算构成群时，必须逐一验证这些公理。注意交换律不是必须的；若群中任意元素满足 a*b=b*a，则称为阿贝尔群（后面会详述）。
3. Basic Properties of Groups | 群的基本性质
From the group axioms, several fundamental properties can be deduced. These facts are frequently used in proofs and problem solving.
由群公理可以推导出若干基本性质，这些结论常在证明与解题中使用。
Uniqueness of identity: The identity element e is unique. If e’ is also an identity, then e = e * e’ = e’.
单位元的唯一性：单位元 e 是唯一的。若 e’ 也是单位元，则 e = e * e’ = e’。
Uniqueness of inverses: For each a ∈ G, the inverse a⁻¹ is unique. If both b and c are inverses of a, then b = b * e = b * (a * c) = (b * a) * c = e * c = c.
逆元的唯一性：对每个 a∈G，其逆元 a⁻¹ 唯一。若 b 和 c 都是 a 的逆元，则推出 b=c。
Reversal law: For any a, b ∈ G, (a * b)⁻¹ = b⁻¹ * a⁻¹. This extends to products of more elements.
逆元反转律：对任意 a,b∈G，(a*b)⁻¹ = b⁻¹*a⁻¹。这一规律可推广到多个元素的乘积。
Cancellation laws: If a * b = a * c, then b = c (left cancellation). Similarly, b * a = c * a implies b = c (right cancellation).
消去律：若 a*b = a*c，则 b=c（左消去）；同理 b*a = c*a 推出 b=c（右消去）。
4. Examples of Groups | 群的例子
Familiarity with standard groups is crucial for the exam. The table below summarises some important groups that appear frequently in Edexcel questions.
熟悉标准群对考试至关重要。下表总结了 Edexcel 试题中经常出现的重要群。
| Group (English) | 群 (中文) | Set | Operation | Abelian? |
|---|---|---|---|---|
| Integers under addition | 整数加法群 | ℤ | + | Yes |
| Non-zero real numbers under multiplication | 非零实数乘法群 | ℝ\{0} | × | Yes |
| Complex roots of unity (e.g., cube roots {1, ω, ω²}) | 复单位根群 | {1, ω, ω²} | × | Yes |
| Integers modulo n under addition | 模 n 加法群 ℤ₍ | {0,1,…,n−1} | + mod n | Yes |
| Symmetric group S₃ | 对称群 S₃ | Permutations of {1,2,3} | Composition | No |
| Klein four-group V | 克莱因四元群 | {e, a, b, c} | * (see Cayley table) | Yes |
Note: the symmetric group Sₙ for n ≥ 3 is non-Abelian. The Klein four-group is the smallest non-cyclic group.
注意：当 n≥3 时对称群 Sₙ 是非阿贝尔群。克莱因四元群是最小的非循环群。
5. 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 groups encountered in basic algebra are Abelian, but matrix groups and permutation groups often are not.
如果对所有 a,b∈G 都有 a*b = b*a，则群 (G, *) 称为阿贝尔群（或交换群）。基础代数中许多群都是阿贝尔群，但矩阵群和置换群常不是。
Examples of Abelian groups: (ℤ, +), (ℝ\{0}, ×), (ℤₙ, + mod n), the Klein four-group V.
阿贝尔群的例子：(ℤ,+)，(ℝ\{0},×)，(ℤₙ, 模 n 加法)，克莱因四元群 V。
Non-Abelian examples: The symmetric group S₃, the general linear group GL(2, ℝ) under matrix multiplication, and the dihedral groups Dₙ for n ≥ 3.
非阿贝尔群例子：对称群 S₃，一般线性群 GL(2,ℝ) 在矩阵乘法下，以及二面体群 Dₙ (n≥3)。
In the exam, you may be asked to prove a group is Abelian by showing that a particular relation holds, or to deduce properties from a Cayley table that is symmetric about the main diagonal.
考试中可能要求通过证明特定关系成立来说明一个群是阿贝尔群，或者从凯莱表关于主对角线对称来推断其为阿贝尔群。
6. Subgroups | 子群
A subset H of a group G is a subgroup if H itself forms a group under the same operation as G. This is denoted H ≤ G. To prove H is a subgroup, you can use the subgroup test:
群 G 的子集 H 若在 G 的运算下自身构成群，则称 H 为子群，记作 H≤G。要证明 H 是子群，可使用子群检验：
H is non-empty, and for all a, b ∈ H, a * b⁻¹ ∈ H. (This one-step test combines closure and inverses.)
H 非空，且对所有 a,b∈H 都有 a*b⁻¹∈H。（这个一步检验涵盖了封闭性和逆元。）
Alternatively, you can check: (i) e ∈ H, (ii) H closed under *, (iii) for each a ∈ H, a⁻¹ ∈ H.
也可以分步检验：(i) e∈H，(ii) H 在 * 下封闭，(iii) 每个 a∈H 的逆元也在 H 中。
Common examples: the set of even integers 2ℤ is a subgroup of (ℤ,+); the set {1, −1} is a subgroup of (ℝ\{0},×); in ℤ₆ the subset {0,2,4} is a subgroup under addition modulo 6.
常见例子：偶数集 2ℤ 是 (ℤ,+) 的子群；{1,−1} 是 (ℝ\{0},×) 的子群；在 ℤ₆ 中，子集 {0,2,4} 是模 6 加法下的子群。
7. Lagrange’s Theorem | 拉格朗日定理
For a finite group G, the order of any subgroup H (denoted |H|) divides the order of G (|G|). This is one of the most powerful tools in group theory.
对于有限群 G，任意子群 H 的阶（记为 |H|）整除 G 的阶（|G|）。这是群论中最强大的工具之一。
|H| divides |G|
|H| 整除 |G|
Consequences: If |G| = p, a prime, then G has no proper non-trivial subgroups (only {e} and G itself). Also, the order of any element a ∈ G (the smallest positive integer n with aⁿ = e) must divide |G|.
推论：若 |G|=p 为素数，则 G 没有非平凡的真子群（只有 {e} 和 G）。此外，任意元素 a∈G 的阶（使 aⁿ=e 的最小正整数 n）必整除 |G|。
For example, if |G| = 6, possible subgroup orders are 1, 2, 3, and 6. This restricts the search for subgroups.
例如，若 |G|=6，则可能的子群阶为 1,2,3,6，这大大缩小了子群的搜寻范围。
8. Cyclic Groups | 循环群
A group G is cyclic if there exists an element a ∈ G such that every element of G can be written as aⁿ for some integer n. The element a is called a generator, and we write G =
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