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SSC Pure Mathematics QUESTION #1255
Question 1
If \(\varphi: \mathbb{Z}_6 \to \mathbb{Z}_6\) is a one-to-one (injective) homomorphism, then \(\ker(\varphi)\) is:
  • {\(\bar{0}, \bar{2}, \bar{4}\)}
  • {\(\bar{0}, \bar{3}\)}
  • {\(\bar{0}\)}✔️
  • All of \(\mathbb{Z}_6\)
Correct Answer Explanation
A one-to-one (injective) homomorphism has trivial kernel. Since \(\ker(\varphi) = \{x : \varphi(x) = 0\}\) and \(\varphi\) is injective, only \(\varphi(0)=0\), so \(\ker(\varphi) = \{\bar{0}\}\).