Examples of groups
| Symbol | Operation | Identity | Elements | Inverse | Abelian |
|---|---|---|---|---|---|
| addition | 0 | yes | |||
| multiplication | 1 | yes | |||
| addition mod | 0 | yes | |||
| multiplication | 1 | yes | |||
| multiplication | 1 | yes | |||
| matrix multiplication | matrices | by Gaußian elimination | no | ||
| multiplication mod | 1 | by Euclid's algorithm | yes | ||
| componentwise addition | yes | ||||
| matrix multiplication | matrices of determinant 1 | inverse matrix | yes | ||
| composition | no | ||||
| orthogonal rotation | rotation (matrices) | inverse rotation (matrix) |
From Contemporary Abstract Algebra
Particular groups
- Vector space
- Geometry
- Chemistry and crystallography
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