Example: Subgroups of Z8
Let G be the cyclic group Z8 whose elements are
and whose group operation is addition modulo eight. Its Cayley table is
| + | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 |
|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 2 | 4 | 6 | 1 | 3 | 5 | 7 |
| 2 | 2 | 4 | 6 | 0 | 3 | 5 | 7 | 1 |
| 4 | 4 | 6 | 0 | 2 | 5 | 7 | 1 | 3 |
| 6 | 6 | 0 | 2 | 4 | 7 | 1 | 3 | 5 |
| 1 | 1 | 3 | 5 | 7 | 2 | 4 | 6 | 0 |
| 3 | 3 | 5 | 7 | 1 | 4 | 6 | 0 | 2 |
| 5 | 5 | 7 | 1 | 3 | 6 | 0 | 2 | 4 |
| 7 | 7 | 1 | 3 | 5 | 0 | 2 | 4 | 6 |
This group has a pair of nontrivial subgroups: J={0,4} and H={0,2,4,6}, where J is also a subgroup of H. The Cayley table for H is the top-left quadrant of the Cayley table for G. The group G is cyclic, and so are its subgroups. In general, subgroups of cyclic groups are also cyclic.
Read more about this topic: Subgroup