Solution for Section G.2 Question 3

3. The elements in the cyclic subgroup generated by [2] are:
[2],
[2] + [2] = [4],
[2] + [2] + [2] = [6],
[2] + [2] + [2] + [2] = [8] = [0],
so the cyclic subgroup generated by [2] is {[0],[2],[4],[6]} with addition modulo 8.

The elements in the cyclic subgroup generated by [3] are:
[3],
[3] + [3] = [6],
[3] + [3] + [3] = [1],
[3] + [3] + [3] + [3] = [4],
[3] + [3] + [3] + [3] + [3] = [7],
[3] + [3] + [3] + [3] + [3] + [3] = [2],
[3] + [3] + [3] + [3] + [3] + [3] + [3] = [5],
[3] + [3] + [3] + [3] + [3] + [3] + [3] + [3] = [0],
so the cyclic subgroup generated by [3] is the group Z₈ with addition modulo 8.

The order of [2] is 4.
The order of [3] is 8.

[3] is a generator of the group Z₈ with addition modulo 8.

Back to Section G.2