Help Correct My Solution: defining a homomorphism between U(18) and itself

I was asked the following question:

"Find a homomorphism F:U(18) -> U(18) that has kernel(F)={1,17} and which fulfills F(13)=7."

Note that U(18)={1,5,7,11,13,17} under multiplication mod(18).

I was told that my solution, as follows, was incorrect because i had **insufficient proof**. I was wondering if anyone could help me fill in the gaps or supply a proof of why this method can not work.

**My Solution:**

As the identity of U(18) is 1, the ker(F)={1,17} implies that our mapping must map 1 :-> 1 and 17 :-> 1. We are also given 13 :-> 7. So we have an incomplete mapping that looks like this:

F: U(18) -> U(18) s.t.

1 :-> 1

5 :-> ?

7 :-> ?

11 :-> ?

13 :-> 7

17 :-> 1

To figure out where element 5 should map to, note that;

5= 13*17 mod(18)

So,

F(5) = F(13*17)

If we are to describe this mapping so as to be a homomorphism, we must have;

F(13*17) = F(13)*F(17)

However, by the mappings we already have, we know that;

F(13)*F(17) = 7*1 = 7

Therefore, we must have;

F(5) = F(13*17) = F(13)*F(17) = 7*1 = 7.

By the same logic, note that;

11 = 5*13 mod(18)

=> F(11) = F(5*13) = F(5)*F(13) = 7*7 = 49 mod(18) = 13

By the same logic again, note that;

7 = 11*17 mod(18)

=> F(7) = F(11*17) = F(11)*F(17) = 13*1 = 13

Therefore, we have a piecewise function F that is a homomorphism by how we defined it, that looks like this;

F: U(18) -> U(18) s.t.

1 :-> 1

5 :-> 7

7 :-> 13

11 :-> 13

13 :-> 7

17 :-> 1

This mapping, consequently, is the mapping F: U(18) -> U(18) defined as x :-> x^2.

Q.E.D.

WHERE DID I GO WRONG!!!!!!?????!?!????