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?