Subgroup - Example: Subgroups of Z8

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.

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