[[Group]]
# Examples of groups

| Symbol                  | Operation              | Identity | Elements                  | Inverse                    | Abelian |
| ----------------------- | ---------------------- | -------- | ------------------------- | -------------------------- | ------- |
| $\mathbb Z$             | addition               | 0        | $k$                       | $-k$                       | yes     |
| $\mathbb{Q}^+$          | multiplication         | 1        | $m/n$                     | $n/m$                      | yes     |
| $\mathbb{Z}_{n}$        | addition mod $n$       | 0        | $k$                       | $n-k$                      | yes     |
| $\mathbb{R}^\star$      | multiplication         | 1        | $x$                       | $x^{-1}$                   | yes     |
| $\mathbb{C}^\star$      | multiplication         | 1        | $a + bi$                  | $(a+bi)^{-1}$              | yes     |
| $GL(2,F)$               | matrix multiplication  | $I$      | matrices                  | by [[Gaußian elimination]] | no      |
| $\mathbb{Z}_{n}^\times$ | multiplication mod $n$ | 1        | $k$ coprime with $n$      | by [[Euclid's algorithm]]  | yes     |
| $\mathbb{R}^n$          | componentwise addition | $\vab 0$ | $\vab v$                  | $-\vab v$                  | yes     |
| $SL(2,F)$               | matrix multiplication  | $I$      | matrices of determinant 1 | inverse matrix             | yes     |
| $D_{n}$                 | composition            | $R_{0}$  | $R_{\alpha}, L$           | $R_{2\pi-\alpha}, L$       | no      |
| $SO(3)$                 | orthogonal rotation    | $I$      | rotation (matrices)       | inverse rotation (matrix)  |         |

From [[@gallianContemporaryAbstractAlgebra2017|Contemporary Abstract Algebra]]


## Particular groups

- Vector space
  - [[General linear group]]
  - [[Special linear group]]
- Geometry
  - [[Euclidean group]]
- Chemistry and crystallography
  - [[Crystallographic group]]


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