开集#
\(\mathbb{R}\)的子集\(U\)称为开集, 如果它内部的每一点都能被一个完全落在\(U\)内的开区间包住:
$$\forall x \in U,\ \exists \varepsilon > 0,\ \text{使得}\ (x - \varepsilon,\ x + \varepsilon) \subseteq U$$
线性映射#
设\(V, W\)是同一域\(\mathbb{F}\)上的向量空间. 映射\(\varphi: V \to W\)称为线性映射, 如果保持加法与数乘:
$$\varphi(c_1 v_1 + c_2 v_2) = c_1 \varphi(v_1) + c_2 \varphi(v_2),\quad \forall v_1, v_2 \in V,\ c_1, c_2 \in \mathbb{F}$$
同构#
同态#
映射\(\varphi: G \to H\)称为同态 (homomorphism), 如果保持运算:
$$\varphi(g_1 g_2) = \varphi(g_1) \varphi(g_2),\quad \forall g_1, g_2 \in G$$
由此自动得\(\varphi(e_G) = e_H\), \(\varphi(g^{-1}) = \varphi(g)^{-1}\).
同构#
同态\(\varphi: G \to H\)称为同构 (isomorphism), 如果它是双射. 此时存在逆同态\(\varphi^{-1}: H \to G\), 满足\(\varphi^{-1} \circ \varphi = \operatorname{id}_G\), \(\varphi \circ \varphi^{-1} = \operatorname{id}_H\). 记作\(G \cong H\).
直观上, \(G \cong H\)意味着两个群作为代数结构完全相同, 仅元素的标签不同.
示例#
- \(\mathbb{R}/\mathbb{Z} \cong S^1\): \([x] \mapsto e^{2\pi i x}\)是同构
- \((\mathbb{R}, +) \cong (\mathbb{R}_{>0}, \cdot)\): \(x \mapsto e^x\)将加法变成乘法
- \(\mathbb{Z}/6\mathbb{Z} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}\) (中国剩余定理)
商群#
商群是把群\(G\)按某个正规子群\(N\)粘合得到的新群.
正规子群#
子群\(N \leq G\)称为正规子群, 记作\(N \triangleleft G\), 如果:
$$gNg^{-1} = N,\quad \forall g \in G$$
等价地, 左陪集等于右陪集: \(gN = Ng\). Abel 群里每个子群自动正规, 此条件只在非交换群中非平凡.
核与像#
设\(\varphi: G \to H\)是群同态.
核是映射到\(H\)的单位元\(e_H\)的元素全体:
$$\operatorname{Ker}\varphi = \{g \in G : \varphi(g) = e_H\}$$
像是\(\varphi\)实际产出的元素全体:
$$\operatorname{Im}\varphi = \{\varphi(g) : g \in G\}$$
由同态条件易得\(\operatorname{Ker}\varphi\)是\(G\)的正规子群, \(\operatorname{Im}\varphi\)是\(H\)的子群.
商群#
元素\(g \in G\)对应陪集\(gN\), 商群是所有陪集的集合:
$$G/N = \{gN : g \in G\}$$
群运算定义为:
$$(g_1 N) \cdot (g_2 N) = (g_1 g_2) N$$
运算良定义要求\(N\)正规: 设\(g_1’ = g_1 n_1\), \(g_2’ = g_2 n_2\), 则
$$g_1’ g_2’ = g_1 n_1 g_2 n_2 = g_1 g_2 \underbrace{(g_2^{-1} n_1 g_2)}_{\in N} n_2 \in g_1 g_2 N$$
需要\(g_2^{-1} N g_2 \subseteq N\), 这正是正规性的来源.
第一同构定理#
群同态\(\varphi: G \to H\)满足:
$$G / \operatorname{Ker}\varphi \cong \operatorname{Im}\varphi$$
证明: 记\(K = \operatorname{Ker}\varphi\), 定义
$$\overline\varphi: G/K \to \operatorname{Im}\varphi,\quad [g] \mapsto \varphi(g)$$
- 良定义: \([g_1] = [g_2] \iff g_2 = g_1 k,\ k \in K \iff \varphi(g_1) = \varphi(g_2)\)
- 同态: \(\overline\varphi([g_1][g_2]) = \varphi(g_1 g_2) = \varphi(g_1) \varphi(g_2) = \overline\varphi([g_1]) \overline\varphi([g_2])\)
- 单射: \(\overline\varphi([g_1]) = \overline\varphi([g_2]) \Rightarrow \varphi(g_1) = \varphi(g_2) \Rightarrow [g_1] = [g_2]\)
- 满射: \(h = \varphi(g) = \overline\varphi([g])\)
故\(\overline\varphi\)是群同构.
示例#
- \(\mathbb{Z}/n\mathbb{Z}\): 模\(n\)的整数加法群
- \(\mathbb{R}/\mathbb{Z} \cong S^1\): 实数粘合整数, 得到圆周
- \(\mathbb{R}/2\pi\mathbb{Z} \cong S^1\): 角度空间
商环#
商环是把环\(R\)按某个理想\(I\)粘合得到的新环, 思想与商群一致, 但需要更强的子结构.
理想#
子集\(I \subseteq R\)称为 (双边) 理想, 如果:
- \((I, +)\)是\((R, +)\)的子群
- 吸收性: \(\forall r \in R,\ a \in I\), 有\(ra \in I\)且\(ar \in I\)
例如\(n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}\)是\(\mathbb{Z}\)的理想; \((x) = \{x p(x) : p \in \mathbb{R}[x]\}\)是\(\mathbb{R}[x]\)的理想.
商环#
商环是所有陪集的集合:
$$R/I = \{a + I : a \in R\}$$
加法和乘法定义为:
$$[a] + [b] = [a + b],\qquad [a] \cdot [b] = [ab]$$
乘法良定义要求\(I\)是理想: 设\(a’ = a + i\), \(b’ = b + j\), 则
$$a’b’ = ab + \underbrace{aj + ib + ij}_{\in I}$$
第一同构定理#
环同态\(\varphi: R \to S\)满足:
$$R / \operatorname{Ker}\varphi \cong \operatorname{Im}\varphi$$
证明: 记\(I = \operatorname{Ker}\varphi\), 它是\(R\)的理想, 定义
$$\overline\varphi: R/I \to \operatorname{Im}\varphi,\quad [a] \mapsto \varphi(a)$$
- 良定义: \([a_1] = [a_2] \iff a_1 - a_2 \in I \iff \varphi(a_1) = \varphi(a_2)\)
- 保加法: \(\overline\varphi([a_1] + [a_2]) = \varphi(a_1 + a_2) = \varphi(a_1) + \varphi(a_2)\)
- 保乘法: \(\overline\varphi([a_1] [a_2]) = \varphi(a_1 a_2) = \varphi(a_1) \varphi(a_2)\)
- 单射: \(\overline\varphi([a_1]) = \overline\varphi([a_2]) \Rightarrow \varphi(a_1) = \varphi(a_2) \Rightarrow [a_1] = [a_2]\)
- 满射: \(h = \varphi(a) = \overline\varphi([a])\)
故\(\overline\varphi\)是环同构.
示例#
- \(\mathbb{Z}/n\mathbb{Z}\): 模\(n\)的整数环, 当\(n\)为素数时是域
- \(\mathbb{R}[x]/(x^2 + 1) \cong \mathbb{C}\): 添加关系\(x^2 = -1\), \([x]\)扮演\(i\)
- \(\mathbb{R}[x]/(x - a) \cong \mathbb{R}\): 对应"在\(x = a\)处求值"
- \(C^\infty(U)/I_p \cong \mathbb{R}\), 其中\(I_p = \{f : f(p) = 0\}\): 把函数代换成它在\(p\)处的值