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, +
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