Question

If \(f:G\to G'\) is an onto homomorphism with kernel \(K\), then

• A. \(\dfrac{G}{K}\cong G'\)
• B. \(\dfrac{K}{G}\cong G\)
• C. \(\dfrac{K}{G}\cong K\)
• D. \(\dfrac{G}{K}\cong K\)

Hint:

First Isomorphism Theorem: \(G/\ker f \cong \operatorname{Im}f\). If \(f\) is onto, then \(\operatorname{Im}f=G'\).

Answer 1

The Correct Option is A

Concept:
This is a direct application of the First Isomorphism Theorem. If \[ f:G\to G' \] is a group homomorphism, then \[ \frac{G}{\ker f}\cong \operatorname{Im} f \]

Step 1: Identify the kernel. The kernel of \(f\) is given as \[ K \] So, \[ \ker f=K \]

Step 2: Use the fact that \(f\) is onto. Since \(f\) is onto, \[ \operatorname{Im} f=G' \]

Step 3: Apply First Isomorphism Theorem. \[ \frac{G}{\ker f}\cong \operatorname{Im} f \] Substitute \[ \ker f=K \] and \[ \operatorname{Im} f=G' \] Thus, \[ \frac{G}{K}\cong G' \]

Step 4: Final answer. \[ \boxed{\frac{G}{K}\cong G'} \]