math / Structures / free
Abstract Algebra
Compare different systems by the operations and symmetries they share.
Follow the mechanism1 / 3
A set comes with a combination rule. The same collection of symbols can behave differently when the operation changes, so the rule belongs in the definition of the structure.
finite operationZ/6Z · + · selected 3
Focus the idea
Change the operation
| + | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| 0 | 0 | 1 | 2 | 3 | 4 | 5 |
| 1 | 1 | 2 | 3 | 4 | 5 | 0 |
| 2 | 2 | 3 | 4 | 5 | 0 | 1 |
| 3 | 3 | 4 | 5 | 0 | 1 | 2 |
| 4 | 4 | 5 | 0 | 1 | 2 | 3 |
| 5 | 5 | 0 | 1 | 2 | 3 | 4 |
Inspect element
closureyes
associativeyes
identity0
structuregroup
Identity0
Leaves every element unchanged.
Inverse of 33
3 + 3 returns to 0.
Every element invertible?yes
All rows have a partner for the identity.
With a + b (mod 6), closure, associativity, identity 0, and an inverse for every element all hold. The set and operation form a group.
Abstract algebra studies the rules that make a system coherent. The labels can change:from numbers to symmetries or matrices:but the same questions remain: can you combine objects, is there a neutral element, and can every move be undone?