Questions in category: 群论 (Group Theory)

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## 1. 证明 $\mathbb{Z}$ 的全部子群为 $\{m\mathbb{Z}\mid m\in\mathbb{Z}\}$.

Posted by haifeng on 2024-01-10 11:37:34 last update 2024-01-10 11:37:34 | Answers (1) | 收藏

## 2. 模群(modular group)

Posted by haifeng on 2022-11-21 14:44:28 last update 2022-11-26 11:08:53 | Answers (1) | 收藏

## 3. 朗兰兹对偶群(Langlands dual group)

Posted by haifeng on 2022-07-26 11:02:28 last update 2022-07-26 11:02:28 | Answers (0) | 收藏

## 4. Weil group(韦依群)

Posted by haifeng on 2022-07-26 11:01:02 last update 2022-07-26 11:02:04 | Answers (0) | 收藏

$\mathbb{R}$, $\mathbb{C}$ 的 Weil 群(Weil group) 分别指 $W_{\mathbb{R}}$, $W_{\mathbb{C}}$.

$W_{\mathbb{C}}=\mathbb{C}^{\times}$.

## 5. 求正有理数构成的乘法群的自同构群.

Posted by haifeng on 2022-06-07 08:35:33 last update 2022-06-07 08:36:35 | Answers (0) | 收藏

Glass 和 Ribenboim [1]证明了 $(\mathbb{Q}^+,\cdot)$ 的保序自同构是平凡自同构.

Theorem.  The only automorphism of the ordered multiplicative group of positive rational numbers is the trivial automorphism.

References:

[1]  A. M. W. Glass and Paulo Ribenboim, Automorphisms of the ordered multiplicative group of positive rational numbers, Proceedings of the American Mathematical Society, (1994) Vol.122, No. 1, 15--18.

## 6. [Def]单群

Posted by haifeng on 2022-03-20 20:12:28 last update 2024-01-10 09:52:03 | Answers (0) | 收藏

Remark: 若称 $\{e\}$ 和 $G$ 是群 $G$ 的平凡正规子群, 则单群就是指没有非平凡的正规子群的群.

Remark: 单群的定义与素数的定义有相似之处.

## 7. 正二十面体变到自身的所有旋转构成一个群, 证明这个群由60个旋转组成.

Posted by haifeng on 2022-01-19 23:08:44 last update 2022-01-19 23:11:10 | Answers (0) | 收藏

References:

[1] Milnor, The Poincaré conjecture.  http://www.claymath.org/sites/default/files/poincare.pdf

## 8. 群的自由乘积

Posted by haifeng on 2020-07-11 19:48:06 last update 2020-07-11 20:10:19 | Answers (1) | 收藏

$*_{\lambda\in\Lambda}G_{\lambda}=\{x_1 x_2\cdots x_n\mid n\geqslant 0, x_i\ \text{是某个}\ G_{\lambda}\ \text{中的非单位元,}\ \text{若}\ i\neq j, \text{则}\ x_i\ \text{与}\ x_{j}\ \text{不在同一个}\ G_{\lambda}\ \text{中}\}$

$(x_1 x_2\cdots x_n)\cdot(y_1 y_2\cdots y_m)= \begin{cases} x_1\cdots (x_{n-\ell}y_{\ell+1})\cdots y_m, & \text{若}\ x_{n-\ell}, y_{\ell+1}\ \text{同属于某个群}\ G_{\lambda}\ \text{中},\\ x_1\cdots x_{n-\ell}y_{\ell+1}\cdots y_m, & \text{否则}. \end{cases}$

Prop. 设 $H$ 是一群, $\forall\ \lambda\in\Lambda$, 有同态 $f_{\lambda}:\ G_{\lambda}\rightarrow H$, 则存在唯一同态 $f:\ *_{\lambda\in\Lambda}G_{\lambda}\rightarrow H$, 使得 $f\bigr|_{G_{\lambda}}=f_{\lambda}$, $\forall\ \lambda\in\Lambda$.

## 9. 设 $H\leqslant G$, 且 $[G:H]=2$, 证明 $H\unlhd G$.

Posted by haifeng on 2018-01-17 21:55:11 last update 2018-01-17 21:55:11 | Answers (0) | 收藏

## 10. $G\times G'$

Posted by haifeng on 2015-02-11 15:55:32 last update 2015-02-11 15:58:20 | Answers (1) | 收藏

$(g,g')\cdot(h,h')=(gh,g'h'),$

\begin{aligned} G\times G'/G\times{e'}\cong G',\\ G\times G'/\{e\}\times G'\cong G.\\ \end{aligned}

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