Problem

Let be the ring of integers modulo , and let be the group of all invertible elements in under multiplication. Let be a group homomorphism with kernel If , which of the following elements is also mapped to 7 under ? (A) 11 (B) 13 (C) 17 (D) 19 (E) 29

---

🔧 Solution Strategy

We use the coset property of kernels for group homomorphisms:

For any group homomorphism , the full preimage (or fiber) of an element is the left coset:

So here, since , the set of all elements in that map to 7 is:

Now compute:

✅ So as well.

---

🧠 Final Answer:

---

💡 Group Theory Takeaway

If you know:

• the image of an element ,
• and the kernel of a homomorphism ,

then you can immediately determine all other preimages of using cosets of the kernel.

---

Hint (click to expand)

Use the fact that for a group homomorphism , the preimage of is the coset .