##### 陪集 - 陪集 - **陪集**是[[群]]的[[子群]]与其元素通过群运算生成的[[子集]], 构成群的[[集合划分|划分]]. 陪集分为左陪集和右陪集, 左右陪集相等当且仅当子群为[[正规子群]]. 设 $H$ 是群 $G$ 的子群, $g\in G$ 为任意元素, 若 $G$ 是[[交换群]], 则左陪集和右陪集相同. **子群指数** $[G:H]$ 定义为 $G$ 中以 $H$ 为基准的陪集个数, 满足[[拉格朗日定理]] - 左陪集, $gH=\{gh\mid h\in H\}$ - 右陪集, $Hg=\{hg\mid h\in H\}$ - 若 $G$ 是[[有限群]], $[G:H]=\frac{|G|}{|H|}$ - 若 $G$ 是[[无限群]], $[G:H]$ 是陪集的基数 >[!example]- 陪集 >- 模 $6$ 加法群 $G = \mathbb{Z}/6\mathbb{Z} = \{ 0, 1, 2, 3, 4, 5 \}$ > - 子群 $H = \{ 0, 3 \}$, 交换群左陪集和右陪集相同 > - $0 + H = \{ 0 + 0, 0 + 3 \} = \{ 0, 3 \}$ > - $1 + H = \{ 1 + 0, 1 + 3 \} = \{ 1, 4 \}$ > - $2 + H = \{ 2 + 0, 2 + 3 \} = \{ 2, 5 \}$ > - $3 + H = \{ 3 + 0, 3 + 3 \} = \{ 3, 6 \equiv 0 \} = \{ 0, 3 \}$ > - $4 + H = \{ 4 + 0, 4 + 3 \} = \{ 4, 7 \equiv 1 \} = \{ 1, 4 \}$ > - $5 + H = \{ 5 + 0, 5 + 3 \} = \{ 5, 8 \equiv 2 \} = \{ 2, 5 \}$ > - 所有不同陪集为 > - $\{ 0, 3 \}$ > - $\{ 1, 4 \}$ > - $\{ 2, 5 \}$ > - 陪集个数为 > - $|G|/|H| = 6/2 = 3$ >- 对称群 $S_3 = \{ e, (12), (13), (23), (123), (132) \}$ > - 子群 $H = \{ e, (12) \}$ > - 左陪集 > - $eH = \{ e \circ e, e \circ (12) \} = \{ e, (12) \}$ > - $(12)H = \{ (12) \circ e, (12) \circ (12) \} = \{ (12), e \} = \{ e, (12) \}$ > - $(13)H = \{ (13) \circ e, (13) \circ (12) \} = \{ (13), (13)(12) = (132) \}$ > - $(23)H = \{ (23) \circ e, (23) \circ (12) \} = \{ (23), (23)(12) = (123) \}$ > - $(123)H = \{ (123) \circ e, (123) \circ (12) \} = \{ (123), (123)(12) = (23) \} = \{ (123), (23) \}$ > - $(132)H = \{ (132) \circ e, (132) \circ (12) \} = \{ (132), (132)(12) = (13) \} = \{ (132), (13) \}$ > - 不同左陪集为 > - $\{ e, (12) \}$ > - $\{ (13), (132) \}$ > - $\{ (23), (123) \}$ > - 右陪集 > - $He = \{ e \circ e, (12) \circ e \} = \{ e, (12) \}$ > - $H(12) = \{ e \circ (12), (12) \circ (12) \} = \{ (12), e \} = \{ e, (12) \}$ > - $H(13) = \{ e \circ (13), (12) \circ (13) \} = \{ (13), (12)(13) = (123) \}$ > - $H(23) = \{ e \circ (23), (12) \circ (23) \} = \{ (23), (12)(23) = (132) \}$ > - $H(123) = \{ e \circ (123), (12) \circ (123) \} = \{ (123), (12)(123) = (13) \}$ > - $H(132) = \{ e \circ (132), (12) \circ (132) \} = \{ (132), (12)(132) = (23) \}$ > - 不同右陪集为 > - $\{ e, (12) \}$ > - $\{ (13), (123) \}$ > - $\{ (23), (132) \}$ > - 陪集个数 > - $|S_3|/|H| = 6/2 = 3$ > - 左陪集和右陪集不同 > - $H = \{ e, (12) \}$ 不是正规子群