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Math
Emeritus
Thanks
PF Gold
P: 39,552
Yes, you are confusing yourself!.
It does mean that <a> is G itself. That is G consists of the elements, a, a<sup>2</sup>, a<sup>3</sup>, a<sup>3</sup>, a<sup>4</sup>, ..., a<sup>21</sup> (and a<sup>21</sup>= e).
Okay, the subgroup generated by a<sup>7</sup> is, as you said, the subgroup of its powers: a<sup>7</sup>, (a<sup>7</sup>)<sup>2</sup>= a<sup>14</sup>, (a<sup>7</sup>)<sup>3</sup>= a<sup>21</sup>= u. How many members does that subgroup have?

Now do the same thing with a<sup>3</sup>. How many powers can you take until you get to a<sup>21</sup>= u?

xy= (a<sup>7</sup>)(a<sup>3</sup>)= a<sup>10</sup>. Same question as above.

I'll let you think longer about part ii.

 Let G be a group, and let x be and element of G. Assume that x has infinite order. Prove that every non-zero power of x also has infinite order. That is, prove that, if i does not equal 0, then xi has infinite order. (Hint: This is a simple proof by contradiction, What happens if (xi)j = u for some positive integer j?)
Continuation of hint: (x<sup>i</sup>)<sup>j</sup>= x<sup>i+j</sup>!