📚 IB WJEC Maths: Introduction to Group Theory – Key Exam Points | IB WJEC 数学：群论入门考点精讲
Group theory is a fundamental branch of abstract algebra and a core topic in the IB WJEC Maths curriculum. It provides a formal framework for studying symmetry and algebraic structures. Mastering the definition of a group, recognising common examples, and understanding key theorems like Lagrange’s Theorem are essential for exam success. This revision guide covers all the must-know concepts in a bilingual, point-by-point format.
群论是抽象代数的基础分支，也是 IB WJEC 数学课程的核心内容。它为研究对称性和代数结构提供了形式化框架。掌握群的定义、识别常见例子以及理解拉格朗日定理等关键定理对于考试成功至关重要。这份考点精讲以中英双语、逐点对照的形式，涵盖了所有必知概念。
1. What is a Group? | 什么是群？
A group (G, *) consists of a non-empty set G and a binary operation * that satisfies four axioms: closure, associativity, identity and inverses.
一个群 (G, *) 由一个非空集合 G 和一个二元运算 * 组成，并且必须满足四条公理：封闭性、结合律、单位元存在性和逆元存在性。
Closure: For all a, b in G, the result of the operation a * b is also in G.
封闭性：对于 G 中所有的 a, b，运算结果 a * b 依然属于 G。
Associativity: For all a, b, c in G, (a * b) * c = a * (b * c).
结合律：对于 G 中所有的 a, b, c，有 (a * b) * c = a * (b * c)。
Identity: There exists an element e in G such that for every a in G, e * a = a * e = a.
单位元：存在元素 e ∈ G，使得对每个 a ∈ G，满足 e * a = a * e = a。
Inverses: For each a in G, there exists an element a⁻¹ in G such that a * a⁻¹ = a⁻¹ * a = e.
逆元：对每个 a ∈ G，都存在 a⁻¹ ∈ G，使得 a * a⁻¹ = a⁻¹ * a = e。
These four axioms must be checked when proving a set with an operation forms a group. Common mistakes include forgetting to verify closure or assuming inverses exist without checking.
在证明一个集合和运算构成群时，必须逐一验证这四条公理。常见的错误包括忘记验证封闭性，或是在没有检查的情况下默认逆元存在。
2. Examples of Groups | 群的例子
The set of integers ℤ under addition, (ℤ, +), is a classic group. The identity is 0, and the inverse of n is -n. This group is infinite and abelian.
整数集 ℤ 在加法运算下构成群 (ℤ, +)。单位元是 0，n 的逆元是 -n。这个群是无限群，并且是交换群。
The non-zero real numbers ℝ\{0} under multiplication form a group (ℝ\{0}, ×). The identity is 1, and the inverse of x is 1/x. Note that 0 must be excluded because it has no multiplicative inverse.
非零实数集 ℝ\{0} 在乘法运算下构成群 (ℝ\{0}, ×)。单位元是 1，x 的逆元是 1/x。注意必须排除 0，因为 0 没有乘法逆元。
The symmetry group of an equilateral triangle, denoted D₃ or S₃, consists of 6 rigid motions (rotations and reflections). This is an example of a non-abelian group, where the order of operations matters.
正三角形的对称群，记作 D₃ 或 S₃，包含 6 种刚体运动（旋转和反射）。这是一个非交换群的例子，其中运算的顺序会影响结果。
D₃ = {e, r, r², s, rs, r²s} with |D₃| = 6
D₃ = {e, r, r², s, rs, r²s}，群的阶为 6
3. Abelian Groups | 交换群（阿贝尔群）
A group (G, *) is called abelian (or commutative) if for every a, b ∈ G, a * b = b * a. This property simplifies many proofs and calculations.
如果对于群 G 中的任意元素 a 和 b，都有 a * b = b * a，则称该群为交换群或阿贝尔群。这个性质能大大简化许多证明和计算。
All groups of prime order are abelian. Additive groups of numbers are always abelian, whereas groups of matrices under multiplication are often non-abelian.
所有素数阶的群都是交换群。数的加法群总是交换的，而矩阵的乘法群通常是非交换的。
Example: In ℤ₅ under addition mod 5, 2 + 4 ≡ 4 + 2 ≡ 1 (mod 5)
例子：在模 5 加法下的 ℤ₅ 中，2 + 4 ≡ 4 + 2 ≡ 1 (mod 5)
To disprove a group is abelian, find just one pair a, b where a * b ≠ b * a. For instance, in S₃, (12)(13) ≠ (13)(12).
要反证一个群是交换群，只需找到一对元素 a, b 使得 a * b ≠ b * a。例如，在 S₃ 中，(12)(13) ≠ (13)(12)。
4. Order of a Group and 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. If |G| is finite, the group is called a finite group.
群 G 的阶，记作 |G|，是指集合 G 中元素的个数。如果 |G| 是有限的，则称该群为有限群。
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). If no such n exists, the element has infinite order.
群中元素 a 的阶是使得 aⁿ = e 成立的最小正整数 n，其中 aⁿ 表示 a 自我运算 n 次。如果不存在这样的 n，则称该元素具有无限阶。
In ℤ₆ under addition, ord(2) = 3 because 2+2+2 ≡ 0 (mod 6)
在加法群 ℤ₆ 中，ord(2) = 3，因为 2+2+2 ≡ 0 (mod 6)
The identity element always has order 1. In any group, an element and its inverse have the same order.
单位元的阶总是 1。在任何群中，元素和它的逆元具有相同的阶。
5. Subgroups | 子群
A subset H of a group G is a subgroup of G, written H ≤ G, if H itself forms a group under the same operation *. To verify a subset is a subgroup, you can use the subgroup test.
群 G 的子集 H 如果在相同的运算 * 下本身也构成一个群，则称 H 是 G 的子群，记作 H ≤ G。可以利用子群检验来验证一个子集是否为子群。
One-step subgroup test: H is a subgroup of G if H is non-empty and for all a, b ∈ H, a * b⁻¹ ∈ H.
一步子群检验：如果 H 非空，并且对所有 a, b ∈ H，有 a * b⁻¹ ∈ H，那么 H 就是 G 的子群。
Every group has at least two trivial subgroups: {e} and the group G itself. Other subgroups are called proper subgroups.
每个群都至少有两个平凡子群：{e} 和群 G 自身。其他的子群称为真子群。
In ℤ₁₂ under addition, {0, 3, 6, 9} is a subgroup of order 4
在加法群 ℤ₁₂ 中，{0, 3, 6, 9} 是一个 4 阶子群
6. 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 gᵏ for some integer k. The element g is called a generator, and we write G = ⟨g⟩.
如果群 G 中存在一个元素 g，使得 G 的每一个元素都可以表示为 gᵏ（k 为整数），则称 G 为循环群。元素 g 称为生成元，记作 G = ⟨g⟩。
All cyclic groups are abelian. Finite cyclic groups of order n are isomorphic to ℤₙ (the integers modulo n under addition). Infinite cyclic groups are isomorphic to ℤ.
所有的循环群都是交换群。n 阶有限循环群同构于 ℤₙ（在模 n 加法下的整数集）。无限循环群同构于 ℤ。
ℤ₇\{0} under multiplication mod 7 is cyclic: ⟨3⟩ = {3¹,3²,…,3⁶} mod 7
群 ℤ₇\{0} 在模 7 乘法下是循环群：⟨3⟩ = {3¹,3²,…,3⁶} mod 7
The number of generators of a cyclic group of order n is φ(n), where φ is Euler’s totient function.
n 阶循环群的生成元个数是 φ(n)，其中 φ 是欧拉函数。
7. Permutation Groups | 置换群
The symmetric group Sₙ consists of all permutations (bijections) of a set of n elements, with composition as the group operation. Its order is n!.
对称群 Sₙ 由 n 个元素集合上的所有排列（双射）组成，以复合作为群运算。它的阶是 n!。
Permutations are often written in cycle notation. For example, in S₃, the permutation that sends 1 → 2, 2 → 3, 3 → 1 is written as (1 2 3). The identity is (1).
置换常用循环记号表示。例如，在 S₃ 中，将 1 映射到 2，2 映射到 3，3 映射到 1 的置换记作 (1 2 3)。单位置换记作 (1)。
A permutation is even if it can be expressed as an even number of transpositions (2-cycles). The set of all even permutations in Sₙ forms the alternating group Aₙ, a subgroup of index 2.
如果一个置换可以表示为偶数个对换（2-循环）的复合，则称其为偶置换。Sₙ 中所有偶置换构成的集合组成交错群 Aₙ，它是 Sₙ 的一个指数为 2 的子群。
8. Lagrange’s Theorem | 拉格朗日定理
One of the most powerful results in finite group theory is Lagrange’s Theorem: If G is a finite group and H is a subgroup of G, then the order of H divides the order of G.
有限群理论中最强大的结论之一是拉格朗日定理：如果 G 是有限群，H 是 G 的子群，那么 H 的阶整除 G 的阶。
|H| divides |G|
|H| 整除 |G|
A direct consequence is that the order of any element a ∈ G also divides |G|, because the cyclic subgroup ⟨a⟩ generated by a has the same order as the element, and must divide |G|.
该定理的一个直接推论是：群 G 中任意元素 a 的阶也整除 |G|，因为 a 生成的循环子群 ⟨a⟩ 与 a 具有相同的阶，且必须整除 |G|。
To find all possible subgroups of a given finite group, list the divisors of |G|; any subgroup’s order must be one of these. Then test candidates using the subgroup test.
要找出给定有限群的所有可能子群，先列出 |G| 的所有因子；任何子群的阶必须是这些因子之一。然后用子群检验逐一测试候选子集。
Note: The converse of Lagrange’s Theorem is not always true – not every divisor necessarily corresponds to a subgroup.
注意：拉格朗日定理的逆命题并非总成立——并非每个因子都一定对应一个子群。
9. Isomorphism | 同构
Two groups G and H are isomorphic if there exists a bijective function f: G → H that preserves the group structure, i.e., f(a * b) = f(a) • f(b) for all a, b ∈ G. We write G ≅ H.
如果存在一个双射 f: G → H 保持群结构，即对所有 a, b ∈ G，有 f(a * b) = f(a) • f(b)，则称群 G 与 H 同构，记作 G ≅ H。
Isomorphic groups are essentially the same in terms of their algebraic properties. For example, the additive group of even integers 2ℤ is isomorphic to ℤ via the map f(n) = n/2.
同构的群在代数性质上本质是相同的。例如，偶数加群 2ℤ 通过映射 f(n) = n/2 同构于 ℤ。
To prove two groups are not isomorphic, you can compare properties such as: order, existence of elements of a certain order, abelianness, or the number of elements of a given order.
要证明两个群不同构，可以比较它们的性质，如：群的阶、特定阶元素的存在性、是否交换、或给定阶元素的个数。
10. Exam Tips and Common Mistakes | 考试要点与常见错误
Tip 1: Always state clearly which group axioms you are verifying when proving a set with an operation is a group. Show explicit calculations for closure and inverses.
要点 1：在证明一个集合和运算构成群时，务必清晰地逐一说明你所验证的公理。对于封闭性和逆元要展示具体的计算过程。
Tip 2: When working with permutation groups in cycle notation, remember to compose from right to left (unless otherwise specified by your exam board). Be careful with disjoint cycles and their orders.
要点 2：在使用循环记号处理置换群时，记住复合通常是从右向左进行（除非考试局另有规定）。处理不相交的循环及其阶时要格外小心。
Tip 3: For questions involving Lagrange’s Theorem, do not claim a subgroup exists just because the order divides |G|. The theorem only gives a necessary condition, not a sufficient one.
要点 3：对于涉及拉格朗日定理的题目，不要仅因为某个数整除 |G| 就断言存在该阶的子群。该定理只给出必要条件，而不是充分条件。
Common mistake: Forgetting that the identity element must be unique can lead to confusion when proving subgroups. Use the fact that in any group, the identity is the only idempotent element.
常见错误：忘记单位元必须具有唯一性会在证明子群时导致混淆。利用这一事实：在任何群中，单位元是唯一的幂等元。
Common mistake: In a non-abelian group, do not assume (ab)ⁿ = aⁿ bⁿ. This holds only when the group is abelian or when the elements commute. Stick to the definition of the operation.
常见错误：在非交换群中，不要假设 (ab)ⁿ = aⁿ bⁿ。只有当群是交换群或元素可交换时才成立。要紧扣运算的定义。
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