Subgroup generated by a

Theorem 2.2.1 If G is a group an a Є G, then the set of all powers of a forms a cyclicsubgroup of G, called the subgroup generated by a, and denoted .

Theorem 2.2.2 Let G be a group, and let a Є G be an element of finite order t. Then

| |, the size of the subgroup generated by a, is equal to t.

Example 2.2.1 Consider the multiplicative group aЄ Z*19 = {1, 2,... ,18} of order 18.

The group is cyclic, and a generator is g = 2. The subgroups of a Є Z*19, and the

generators are listed in the following table.