解释对称群 $S_n$ 的阶为何等于 $n!$，并说明交错群 $A_n$ 为何是 $S_n$ 的指数为 2 的正规子群。
Explain why the order of the symmetric group $S_n$ is $n!$, and why the alternating group $A_n$ is a normal subgroup of index 2 in $S_n$.
$S_n$ 由 n 个元素的全部置换组成，第一个位置 n 种选择、第二个 n-1 种，依次相乘得 $n!$ 个置换。符号同态 $\operatorname{sgn}:S_n\to\{\pm1\}$ 的核为偶置换集合 $A_n$，其像为二阶群，由同态基本定理 $|S_n|/|A_n|=2$，故 $A_n$ 指数为 2 且为正规子群。
$S_n$ consists of all permutations of n elements; there are n choices for the first position, n-1 for the second, and so on, yielding $n!$ permutations. The kernel of the sign homomorphism $\operatorname{sgn}:S_n\to\{\pm1\}$ is the set $A_n$ of even permutations, whose image has order 2, so by the first isomorphism theorem $|S_n|/|A_n|=2$, making $A_n$ a normal subgroup of index 2.
难度 专业 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《近世代数》刘绍学
Abstract Algebra by Dummit and Foote
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#对称群 #交错群 #置换
范畴论中的「函子」是什么？它如何在群论与拓扑之间建立联系？
What is a functor in category theory, and how does it connect group theory with topology?
函子是从一个范畴到另一个范畴的映射，把对象映为对象、态射映为态射，并保持恒等态射与复合。基本群函子 $\pi_1$ 把拓扑空间映为群、把连续映射映为群同态，从而把拓扑问题转化为代数问题，是代数拓扑的核心工具。
A functor maps one category to another, sending objects to objects and morphisms to morphisms while preserving identity morphisms and composition. The fundamental group functor $\pi_1$ assigns a group to each topological space and a group homomorphism to each continuous map, translating topology into algebra and forming a core tool of algebraic topology.
难度 专业 · 适合 高中(15-18)
CC BY 4.0 · 已核验 · Verified 2026-10-04
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《代数拓扑基础》Hatcher
Algebraic Topology by Allen Hatcher
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780131816299 (2000)
#范畴论 #函子 #代数拓扑
环与域的主要区别是什么？为何有理数集构成域而整数集只构成环？
What is the main difference between a ring and a field? Why do the rationals form a field while the integers form only a ring?
环要求加法成群且乘法满足结合律与分配律，但不要求乘法可逆；域在此基础上还要求非零元素乘法成群。有理数中每个非零数都有乘法逆元，故构成域；整数中如 2 没有整数逆元，因此只是环而非域。
A ring requires addition to form a group and multiplication to be associative and distributive, but not invertible; a field additionally requires nonzero elements to form a multiplicative group. Every nonzero rational has a multiplicative inverse, so the rationals are a field, whereas integers like 2 lack an integer inverse, so they form only a ring.
难度 进阶 · 适合 大众
CC BY 4.0 · 已核验 · Verified 2026-11-02
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《代数学引论》聂灵沼、丁石孙
Algebra by Serge Lang
独立核验依据 · Independent verification: 核验依据 1 ISBN 9787040205251（豆瓣书目条目: 代数学引论）
#环 #域 #代数结构
什么是群同态？举例说明核（kernel）的含义。
What is a group homomorphism? Give an example illustrating the meaning of the kernel.
群同态是两个群之间保持运算的映射 $f:G\to H$，即 $f(ab)=f(a)f(b)$。核 $\ker f$ 是被映到 H 中单位元的 G 的元素集合。例如符号映射从 $S_n$ 到 $\{\pm1\}$，其核正是偶置换群 $A_n$。
A group homomorphism is a map $f:G\to H$ between groups that preserves the operation, so $f(ab)=f(a)f(b)$. The kernel $\ker f$ is the set of elements of G mapped to the identity of H. For example the sign map from $S_n$ to $\{\pm1\}$ has kernel $A_n$, the even permutations.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《近世代数》刘绍学
Abstract Algebra by Dummit and Foote
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#群同态 #核 #映射
什么是群？请用整数加法的例子说明群的四个基本公理。
What is a group? Use integer addition to illustrate the four basic group axioms.
群是带二元运算的集合，满足封闭性、结合律、存在单位元、每个元素有逆元。整数集 $\mathbb{Z}$ 在加法下是群：任意两整数之和仍是整数（封闭），加法满足结合律，0 为单位元，每个整数 $x$ 的逆元是 $-x$。
A group is a set with a binary operation satisfying closure, associativity, an identity element, and an inverse for each element. The integers $\mathbb{Z}$ under addition form a group: the sum of any two integers is an integer (closure), addition is associative, 0 is the identity, and each integer $x$ has inverse $-x$.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《近世代数》刘绍学
Abstract Algebra by Dummit and Foote
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#群 #抽象代数 #公理
什么是阿贝尔群？举一个非阿贝尔群的例子。
What is an abelian group? Give an example of a non-abelian group.
阿贝尔群是运算满足交换律的群，即对任意 a,b 有 $ab=ba$。整数加法群是阿贝尔群。非阿贝尔群的反例是三个以上元素的对称群 $S_n\ (n\ge3)$，其中置换复合一般不交换，例如对换 $(12)$ 与 $(13)$ 复合顺序不同结果不同。
An abelian group is one whose operation is commutative, so $ab=ba$ for all a,b; the integers under addition are abelian. A non-abelian example is the symmetric group $S_n$ for $n\ge3$, where composition of permutations need not commute, e.g. the transpositions $(12)$ and $(13)$ differ when composed in opposite orders.
难度 进阶 · 适合 初中(12-15)
CC BY 4.0 · 已核验 · Verified 2026-10-04
来源 · Source
《近世代数》刘绍学
Abstract Algebra by Dummit and Foote
独立核验依据 · Independent verification: 核验依据 1 ISBN 9780135693025 (2004)
#阿贝尔群 #交换群 #对称群
什么是群表示？为何研究对称群的表示有助于理解分子振动等物理问题？
What is a group representation, and why does studying representations of symmetric groups help understand physical problems such as molecular vibrations?
群表示是把群元素映为线性空间上可逆矩阵的同态，从而用矩阵运算研究抽象群。对称群的不可约表示可分解分子的振动模式：通过表示的直和分解，能判别哪些振动模式是红外或拉曼活性的，这正是群论在化学与物理中的典型应用。
A group representation is a homomorphism from a group into the invertible matrices on a vector space, letting one study abstract groups via matrix algebra. Irreducible representations of symmetric groups decompose molecular vibration modes; the decomposition reveals which modes are infrared or Raman active, a classic application of group theory in chemistry and physics.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：著录仅给书名，无版次与年份；同一书名常对应多个版次，不臆测具体版本。
《群论在物理问题中的应用》Hamermesh
Group Theory and Its Application to Physical Problems by Hamermesh
#群表示 #对称群 #应用
范畴、函子与自然变换是什么？
What are categories, functors and natural transformations?
范畴由“对象”和它们之间的“态射”组成并满足复合律；函子是保持结构的范畴间映射；自然变换是函子间的“映射的映射”。范畴论为数学提供统一的语言。
A category has objects and morphisms with composition; a functor maps categories preserving structure; a natural transformation maps functors. Category theory unifies mathematics.
难度 进阶 · 适合 高中(15-18)
CC BY 4.0 · 来源待核验 · Source pending review
来源 · Source
出处备注：本题原挂出处与答案内容不符，已撤回；具体书目待补，暂不展示来源。
本题原挂出处与答案内容不符，已撤回；具体书目待补。
Previous citation withdrawn as unsupported for this answer; a specific source is pending.
#范畴论 #范畴