For any element a in G, a group, prove that <a> is a subgroup of C(a).

Proof:

Now { } if So e would be in <a> for the least possible n, implies that <a> is nonempty. If <a> = {e}, then <a> is already a subgroup of C(a), so I assume <a> do not equal to {e}.

Let and be in <a>, and consider

Now , which is in <a>, and , which is in <a> as well.

Furthermore, , so , thus proves <a> is a subgroup of C(a).

Q.E.D.

Is that right?