TIFR-2020-Maths-A: 20

Question

Let $A_{4}$ denote the group of even permutations of $\left \{ 1,2,3,4 \right \}$. Consider the following statements:

1. There exists a surjective group homomorphism $A_{4}\rightarrow \mathbb{Z}/4 \mathbb{Z}$.
2. There exists a surjective group homomorphism $A_{4}\rightarrow \mathbb{Z}/3 \mathbb{Z}$.

Which of the following statements is correct?

• A. (I) is true and (II) is false
• B. (II) is true and (I) is false
• C. (I) and (II) are both true
• D. (I) and (II) are both false