Question

Let \( G \) be a noncyclic group of order 4. Consider the statements I and II: 
I. There is NO injective (one-one) homomorphism from \( G \) to \( \mathbb{Z}_4 \) 
II. There is NO surjective (onto) homomorphism from \( \mathbb{Z}_4 \) to \( G \) 
 

• A. I is true
• B. I is false
• C. II is true
• D. II is false

Hint:

In general, a homomorphism from a cyclic group to a noncyclic group cannot be surjective. Also, a homomorphism from a noncyclic group to a cyclic group cannot be injective.

Answer 1

The Correct Option is A, C

Step 1: Understanding the noncyclic group \( G \).
A noncyclic group of order 4 is isomorphic to \( \mathbb{Z}_2 \times \mathbb{Z}_2 \), which means that it has two elements of order 2 and no element of order 4.
Step 2: Analyzing Statement I.
The statement "There is NO injective homomorphism from \( G \) to \( \mathbb{Z}_4 \)" is true. Since \( G \) is noncyclic and \( \mathbb{Z}_4 \) is cyclic, any homomorphism from a noncyclic group to a cyclic group must have a nontrivial kernel, making it noninjective.
Step 3: Analyzing Statement II.
The statement "There is NO surjective homomorphism from \( \mathbb{Z}_4 \) to \( G \)" is true. Since \( \mathbb{Z}_4 \) is cyclic and \( G \) is noncyclic, no homomorphism from a cyclic group to a noncyclic group can be surjective.
Step 4: Conclusion.
Thus, the correct answer is \( \boxed{(C)} \).