Infinite Groups
Some of the sets with major operators are as
| Operations | A.S. | S.G. | M | G | AG |
|---|---|---|---|---|---|
| N, + | ✓ | ✓ | ✗ | ✗ | ✗ |
| N, - | ✗ | ✗ | ✗ | ✗ | ✗ |
| N, × | ✓ | ✓ | ✓ | ✗ | ✗ |
| N, ÷ | ✗ | ✗ | ✗ | ✗ | ✗ |
| Z, + | ✓ | ✓ | ✓ | ✓ | ✓ |
| Z, - | ✓ | ✗ | ✗ | ✗ | ✗ |
| Z, × | ✓ | ✓ | ✓ | ✗ | ✗ |
| Z, ÷ | ✗ | ✗ | ✗ | ✗ | ✗ |
| R, + | ✓ | ✓ | ✓ | ✓ | ✓ |
| R, - | ✓ | ✗ | ✗ | ✗ | ✗ |
| R, × | ✓ | ✓ | ✓ | ✗ | ✗ |
| R, ÷ | ✗ | ✗ | ✗ | ✗ | ✗ |
| Even, + | ✓ | ✓ | ✓ | ✓ | ✓ |
| Even, × | ✓ | ✓ | ✗ | ✗ | ✗ |
| Odd, + | ✗ | ✗ | ✗ | ✗ | ✗ |
| Odd, × | ✓ | ✓ | ✓ | ✗ | ✗ |
| Matrix, + | ✗ | ✗ | ✗ | ✗ | ✗ |
| Matrix, × | ✓ | ✓ | ✓ | ✗ | ✗ |
Here, if an operation is not algebraic structure, then we will not check it further for group. Similarly, if an operation is not semi-group, then we will not check further for group. Similarly, if an operation is not monoid, then we will not check it further for group. Hence, if an operation is not group, then we will not check it for abelian group.
Hence, the operations which are group and abelian group are as
- (Z, +)
- (R, +)
- Even, +
- Matrix, +
Finite Groups
A group (G, *) is called a finite group if G is a finite set.
Example: The group G = {0, 1, 2, 3, 4, 5} under addition modulo 6 is a finite group as the set G is a finite set.
Order of Group
The order of the group G is the number of elements in the group G. It is denoted by |G|. A group of order 1 has only the identity element, i.e., ({e}, *).
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