# 圓羣

${\displaystyle \mathbb {T} :=\{z\in \mathbb {C} :|z|=1\}}$

${\displaystyle \theta \mapsto z=e^{i\theta }=\cos \theta +i\sin \theta }$

## 同構

### R/Z

${\displaystyle \theta \mapsto e^{i\theta }=\cos \theta +i\sin \theta }$

${\displaystyle e^{i\theta _{1}}e^{i\theta _{2}}=e^{i(\theta _{1}+\theta _{2})}}$

### SO(2)

${\displaystyle e^{i\theta }\leftrightarrow f(e^{i\theta })={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}}$

${\displaystyle f(e^{i\theta _{1}})f(e^{i\theta _{2}})={\begin{bmatrix}\cos \theta _{1}&-\sin \theta _{1}\\\sin \theta _{1}&\cos \theta _{1}\end{bmatrix}}{\begin{bmatrix}\cos \theta _{2}&-\sin \theta _{2}\\\sin \theta _{2}&\cos \theta _{2}\end{bmatrix}}={\begin{bmatrix}\cos(\theta _{1}+\theta _{2})&-\sin(\theta _{1}+\theta _{2})\\\sin(\theta _{1}+\theta _{2})&\cos(\theta _{1}+\theta _{2})\end{bmatrix}}=f(e^{i(\theta _{1}+\theta _{2})})}$

## 性質

${\displaystyle \mathbb {Z} /2\mathbb {Z} }$（2階循環羣）對圓羣有個反轉作用，所以可以考慮半直積${\displaystyle \mathbb {T} \rtimes \mathbb {Z} /2\mathbb {Z} }$，呢個羣其實同${\displaystyle O(2,\mathbb {R} )}$同構[6]，幾何上嘅諗法係，圓嘅所有對稱都可以寫做「反轉或者唔反轉」接住一個旋轉。因為呢個係同${\displaystyle \mathbb {Z} /2\mathbb {Z} }$嘅半直積，所以亦都可以話${\displaystyle O(2,\mathbb {R} )}$${\displaystyle \mathbb {T} }$廣義二面體羣。呢個羣亦都係無限二面體羣嘅連續版本，而任何嘅二面體羣都作為子羣裝咗喺無限二面體羣入面，所以亦都裝喺呢個羣入面。

## 表示理論

${\displaystyle \phi _{n}}$嚟代表以下嘅表示（n整數）：

${\displaystyle \phi _{n}(e^{i\theta })=e^{in\theta }}$

${\displaystyle \phi _{-n}={\overline {\phi _{n}}}}$

${\displaystyle \mathrm {Hom} (\mathbb {T} ,\mathbb {T} )\cong \mathbb {Z} }$

${\displaystyle \rho _{n}(e^{i\theta })={\begin{bmatrix}\cos n\theta &-\sin n\theta \\\sin n\theta &\cos n\theta \end{bmatrix}},n\in \mathbb {Z} ^{+}}$

## 純代數結構

${\displaystyle \mathbb {T} \cong \mathbb {R} \oplus (\mathbb {Q} /\mathbb {Z} )}$

## 其他性質

• 如果${\displaystyle G}$係一個局部緊緻郝斯多夫羣，而且佢每個真閉子羣都只有有限個閉子羣，咁${\displaystyle G}$就係拓樸同構於圓羣。[11]

## 參考

1. James, Robert C.; James, Glenn (1992). Mathematics Dictionary (第Fifth版). Chapman & Hall. p. 436. ISBN 9780412990410. a unit complex number is a complex number of unit absolute value
2. "circle group in nLab". ncatlab.org. 喺2022-05-16搵到.
3. Gowers, Timothy (2010). The Princeton Companion to Mathematics. Princeton University Press.
4. "Maths in three minutes: Groups". Plus Maths (英文). 喺2022-05-21搵到.
5. Samimullah, Miakhel (12-2020). "The Usage of Cyclic Group in the Clock" (PDF). International Journal of Science and Research (IJSR). 9. {{cite journal}}: Check date values in: |date= (help)
6. "Orthogonal group:O(2,R) - Groupprops". groupprops.subwiki.org. 喺2022-05-16搵到.
7. Kirillov, Aleksandr A. (1976). Elements of the Theory of Representations. Grundlehren der mathematischen Wissenschaften (英文).第220卷. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-642-66243-0. ISBN 978-3-642-66245-4.
8. "divisible group in nLab". ncatlab.org. 喺2022-05-16搵到.
9. "Q/Z in nLab". ncatlab.org. 喺2022-05-21搵到.
10. Clay, James R (1969–10). "The punctured plane is isomorphic to the unit circle". Journal of Number Theory (英文). 1 (4): 500–501. doi:10.1016/0022-314X(69)90011-0.{{cite journal}}: CS1 maint: date format (link)
11. Morris, Sidney A. (1987–10). "The circle group". Bulletin of the Australian Mathematical Society (英文). 36 (2): 279–282. doi:10.1017/S000497270002654X. ISSN 0004-9727.{{cite journal}}: CS1 maint: date format (link)

## 註

1. 喺呢篇入邊統一用${\displaystyle \mathbb {T} }$