Group Theory Essentials for IB CCEA Mathematics | IB CCEA 数学：群论入门考点精讲

📚 Group Theory Essentials for IB CCEA Mathematics | IB CCEA 数学：群论入门考点精讲

Group theory is a central branch of abstract algebra that studies algebraic structures known as groups. For IB CCEA Mathematics, understanding the axioms, examples, and key theorems such as Lagrange’s Theorem is essential for success in the optional topic of Sets, Relations and Groups. This article provides a comprehensive yet concise revision of the core concepts and techniques you need to master.

群论是抽象代数的一个核心分支，研究被称为群的代数结构。对于 IB CCEA 数学而言，理解公理、示例以及拉格朗日定理等关键定理，是掌握集合、关系与群这一选修主题的关键。本文将全面而精炼地复习你需要掌握的核心概念和技巧。

1. Group Axioms and Basic Definitions | 群公理与基本概念

A group is a set G together with a binary operation * that satisfies four axioms. We write (G, *) or simply G when the operation is clear.

群是一个集合 G 连同其上的一个二元运算 *，满足四条公理。当运算明确时，我们记作 (G, *) 或简记为 G。

The axioms are: closure (a * b ∈ G for all a, b ∈ G), associativity ((a * b) * c = a * (b * c)), identity (there exists e ∈ G such that e * a = a * e = a for all a), and inverses (for each a ∈ G, there exists a⁻¹ ∈ G such that a * a⁻¹ = a⁻¹ * a = e).

公理为：封闭性（对所有 a, b ∈ G，有 a * b ∈ G）、结合律（(a * b) * c = a * (b * c)）、单位元（存在 e ∈ G，使对所有 a 有 e * a = a * e = a）以及逆元（对每个 a ∈ G，存在 a⁻¹ ∈ G，使得 a * a⁻¹ = a⁻¹ * a = e）。

Closure: ∀ a, b ∈ G, a * b ∈ G

These axioms are fundamental; any structure failing even one is not a group. Commutativity is not required, but if a * b = b * a for all a, b, the group is called abelian.

这些公理是基础；任何结构只要有一条不满足就不是群。交换律不是必需的，但若对所有 a, b 都有 a * b = b * a，则称该群为阿贝尔群。

The identity element e is unique, and each element’s inverse is unique. In additive notation, the identity is 0 and the inverse of a is −a.

单位元 e 是唯一的，每个元素的逆元也是唯一的。在加法记号下，单位元为 0，a 的逆元为 −a。

2. Examples of Groups | 群的示例

The set of integers ℤ under addition forms an abelian group: 0 is the identity, and the inverse of n is −n. Similarly, ℚ, ℝ, and ℂ under addition are groups.

整数集 ℤ 在加法下构成阿贝尔群：0 是单位元，n 的逆元是 −n。类似地，有理数集 ℚ、实数集 ℝ 和复数集 ℂ 在加法下也是群。

The non-zero real numbers ℝ* = ℝ \ {0} under multiplication form a group. The identity is 1, and the inverse of x is 1/x. The positive reals ℝ⁺ also form a multiplicative group.

非零实数 ℝ* = ℝ \ {0} 在乘法下构成群。单位元是 1，x 的逆元是 1/x。正实数 ℝ⁺ 也在乘法下构成群。

The set of complex numbers of modulus 1, {z ∈ ℂ : |z| = 1}, under multiplication is a group (the circle group). Finite groups include the integers modulo n under addition, denoted ℤₙ, with identity 0 and inverse −a mod n.

模为 1 的复数集 {z ∈ ℂ : |z| = 1} 在乘法下是一个群（圆周群）。有限群包括模 n 整数加法群 ℤₙ，单位元是 0，a 的逆元是 −a mod n。

The set of invertible 2×2 matrices with real entries under matrix multiplication, GL(2, ℝ), is a non-abelian group. This illustrates that group elements need not be numbers.

实数域上的可逆 2×2 矩阵在矩阵乘法下构成 GL(2, ℝ) 群，这是一个非阿贝尔群，说明群的元素不一定是数。

3. 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 G. For example, |ℤₙ| = n, and |ℤ| is infinite.

群的阶记作 |G|，是群中元素的个数。例如 |ℤₙ| = n，而 |ℤ| 是无穷。

The order of an element a in G is the smallest positive integer m such that aᵐ = e (in multiplicative notation). If no such m exists, the element has infinite order.

元素的阶是使 aᵐ = e 成立的最小正整数 m（乘法记号下）。若不存在这样的 m，则该元素是无限阶的。

In ℤₙ, the order of an element k is n / gcd(k, n). For example, in ℤ₁₂, the element 8 has order 12 / gcd(8,12) = 12 / 4 = 3.

在 ℤₙ 中，元素 k 的阶为 n / gcd(k, n)。例如在 ℤ₁₂ 中，元素 8 的阶为 12 / gcd(8,12) = 12/4 = 3。

The order of an element always divides the order of the group in the finite case (a consequence of Lagrange’s Theorem). The identity is the unique element of order 1.

在有限群中，元素的阶总是整除群的阶（拉格朗日定理的推论）。单位元是唯一阶为 1 的元素。

4. Subgroups: Criteria and Examples | 子群：判别标准与实例

A subset H of a group G is a subgroup if H is itself a group under the same operation. We write H ≤ G. The trivial subgroups are {e} and G itself.

群 G 的子集 H 若在相同运算下本身构成群，则称 H 为子群，记作 H ≤ G。平凡子群是 {e} 和 G 本身。

The subgroup test: a non-empty subset H ⊆ G is a subgroup if and only if for all a, b ∈ H we have a * b⁻¹ ∈ H. This single condition ensures closure, identity, and inverses.

子群判别法：非空子集 H ⊆ G 是子群当且仅当对所有 a, b ∈ H 有 a * b⁻¹ ∈ H。这一条件保证了封闭性、单位元和逆元。

Examples: The even integers 2ℤ form a subgroup of ℤ under addition. The set {1, −1} is a subgroup of ℝ* under multiplication. The rotation symmetries of a square form a subgroup of the dihedral group D₄.

示例：偶数集 2ℤ 是 ℤ 在加法下的子群。集合 {1, −1} 是 ℝ* 在乘法下的子群。正方形的旋转对称构成二面体群 D₄ 的子群。

If H is finite and non-empty, closure alone suffices: for all a, b ∈ H, a * b ∈ H implies H ≤ G. This is often quicker for small groups.

若 H 是有限非空子集，仅封闭性就足够：对所有 a, b ∈ H 有 a * b ∈ H 即可推出 H ≤ G。这对小群更快捷。

5. 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 is a power of g. We write G = ⟨g⟩ and call g a generator.

若存在元素 g ∈ G 使得 G 中每个元素都是 g 的幂，则 G 是循环群。记作 G = ⟨g⟩，称 g 为生成元。

ℤ under addition is an infinite cyclic group generated by 1 (and also by −1). Every cyclic group of order n is isomorphic to ℤₙ.

加法群 ℤ 是由 1（以及 −1）生成的无限循环群。每个 n 阶循环群都同构于 ℤₙ。

In a finite cyclic group of order n, an element gᵏ is a generator if and only if gcd(k, n) = 1. Thus the number of generators is φ(n), Euler’s totient function.

在 n 阶有限循环群中，元素 gᵏ 是生成元当且仅当 gcd(k, n) = 1。因此生成元的个数为 φ(n)（欧拉函数）。

Subgroups of a cyclic group are also cyclic. For each positive divisor d of n, there is exactly one subgroup of order d, namely ⟨gⁿ/ᵈ⟩.

循环群的子群也都是循环群。对 n 的每个正因子 d，恰好存在一个 d 阶子群，即 ⟨gⁿ/ᵈ⟩。

6. Cosets and Lagrange’s Theorem | 陪集与拉格朗日定理

Let H be a subgroup of G. A left coset of H in G is a set of the form aH = {a * h : h ∈ H} for some a ∈ G. Right cosets are defined analogously as Ha.

设 H 为 G 的子群。左陪集是形如 aH = {a * h : h ∈ H} 的集合（a ∈ G）。类似地可定义右陪集 Ha。

The set of all left cosets partitions G into disjoint subsets of equal size |H|. The number of distinct left cosets is called the index [G : H].

所有左陪集的集合将 G 划分为大小都等于 |H| 的不相交子集，不同左陪集的个数称为指数 [G : H]。

Lagrange’s Theorem: If G is a finite group and H ≤ G, then |H| divides |G|, and |G| = [G : H] × |H|.

拉格朗日定理：若 G 为有限群且 H ≤ G，则 |H| 整除 |G|，且 |G| = [G : H] × |H|。

A direct corollary: the order of any element divides |G|. Hence, in a group of prime order, every non-identity element generates the whole group, making it cyclic and simple.

直接推论：任意元素的阶整除 |G|。因此，素数阶群中每个非单位元都生成整个群，从而该群是循环群且为单群。

For a finite group G and any element a, we have a^|G| = e. This is immediate from Lagrange’s theorem applied to the subgroup generated by a.

对于有限群 G 和任意元素 a，有 a^|G| = e。这可由拉格朗日定理应用于由 a 生成的子群直接得到。

7. Normal Subgroups and Quotient Groups | 正规子群与商群

A subgroup N of G is normal (N ◁ G) if for all g ∈ G and n ∈ N, we have g * n * g⁻¹ ∈ N. Equivalently, left cosets equal right cosets: gN = Ng for all g.

G 的子群 N 称为正规子群（N ◁ G），若对所有 g ∈ G 和 n ∈ N 有 g * n * g⁻¹ ∈ N。等价地，所有左陪集等于右陪集：∀g, gN = Ng。

In an abelian group, every subgroup is normal. For example, all subgroups of ℤₙ are normal. In non-abelian groups, only certain subgroups are normal.

在阿贝尔群中，每个子群都是正规的。例如 ℤₙ 的所有子群都是正规的。在非阿贝尔群中，只有特定的子群是正规的。

If N ◁ G, the set of cosets G/N = {aN : a ∈ G} forms a group under the operation (aN)(bN) = (ab)N. This is the quotient group (or factor group).

若 N ◁ G，则陪集的集合 G/N = {aN : a ∈ G} 在运算 (aN)(bN) = (ab)N 下构成一个群，称为商群（或因子群）。

The identity in G/N is N = eN, and the inverse of aN is a⁻¹N. The order of G/N is the index [G : N] = |G| / |N|.

G/N 中的单位元为 N = eN，aN 的逆元为 a⁻¹N。G/N 的阶为指数 [G : N] = |G| / |N|。

8. Group Homomorphisms and Isomorphisms | 群同态与群同构

A map φ : G → H between two groups is a homomorphism if φ(a * b) = φ(a) ∗ φ(b) for all a, b ∈ G, where ∗ is the operation in H.

若映射 φ : G → H 满足对所有 a, b ∈ G 有 φ(a * b) = φ(a) ∗ φ(b)（∗ 为 H 中运算），则称 φ 为群同态。

Key properties: φ(e_G) = e_H, and φ(a⁻¹) = (φ(a))⁻¹. The image Im(φ) is a subgroup of H, and the kernel Ker(φ) = {g ∈ G : φ(g) = e_H} is a normal subgroup of G.

关键性质：φ(e_G) = e_H，φ(a⁻¹) = (φ(a))⁻¹。像 Im(φ) 是 H 的子群，核 Ker(φ) = {g ∈ G : φ(g) = e_H} 是 G 的正规子群。

A homomorphism is an isomorphism if it is bijective. Then G and H are structurally identical, written G ≅ H. Isomorphisms preserve order, commutativity, and all group-theoretic properties.

若同态是双射，则称为同构，此时 G 和 H 结构完全相同，记作 G ≅ H。同构保持阶、交换性及所有群论性质。

The First Isomorphism Theorem: G / Ker(φ) ≅ Im(φ). This is a powerful tool for simplifying groups and proving their classification.

第一同构定理：G / Ker(φ) ≅ Im(φ)。这是简化群、证明群分类的强有力工具。

9. Cayley Tables and Small Finite Groups | 凯莱表与小阶有限群

A Cayley table displays the group operation for a finite group. Each row and column contains each group element exactly once (Latin square property).

凯莱表展示有限群的运算。每行和每列恰好包含每个群元素一次（拉丁方性质）。

*	e	a	b	c
e	e	a	b	c
a	a	e	c	b
b	b	c	e	a
c	c	b	a	e

Above is the Cayley table for the Klein four-group V₄, an abelian group of order 4 where every element is its own inverse (a² = b² = c² = e). It is isomorphic to ℤ₂ × ℤ₂.

上表是克莱因四元群 V₄ 的凯莱表，这是一个 4 阶阿贝尔群，每个元素都是自身的逆元（a² = b² = c² = e）。它同构于 ℤ₂ × ℤ₂。

Up to isomorphism, there are only two groups of order 4: the cyclic group ℤ₄ and the Klein four-group V₄. Recognizing a group from its Cayley table is an essential skill.

在同构意义下，4 阶群只有两个：循环群 ℤ₄ 和克莱因四元群 V₄。从凯莱表识别群是一项必备技能。

For prime order p, the only group (up to isomorphism) is the cyclic group ℤₚ. Groups of order 6 are either ℤ₆ or the symmetric group S₃ (non-abelian).

对于素数阶 p，唯一的群（同构意义下）是循环群 ℤₚ。6 阶群要么是 ℤ₆，要么是对称群 S₃（非阿贝尔）。

10. Permutation Groups and Cycle Notation | 置换群与循环表示

Let X be a set. The set of all bijections from X to X, denoted Sym(X), forms a group under composition. When X = {1, 2, …, n}, this is the symmetric group Sₙ, of order n!.

设 X 为一集合。X 到自身的所有双射构成的集合 Sym(X) 在复合运算下形成一个群。当 X = {1, 2, …, n} 时，称为对称群 Sₙ，阶为 n!。

A permutation can be written in cycle notation. For example, (1 3 2) means 1→3→2→1. A 2-cycle (i j) is a transposition. Every permutation is a product of disjoint cycles.

置换可用循环记号表示。例如 (1 3 2) 表示 1→3→2→1。2-轮换 (i j) 称为对换。每个置换都可表示为不相交轮换的乘积。

The parity of a permutation (even/odd) is determined by the number of transpositions needed to express it. The set of even permutations forms the alternating group Aₙ, a subgroup of Sₙ of index 2.

置换的奇偶性由表示所需对换的个数决定。全体偶置换构成交错群 Aₙ，它是 Sₙ 的指数为 2 的子群。

Cayley’s Theorem states that every group G is isomorphic to a subgroup of Sym(G). Thus permutation groups serve as universal models for all groups.

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