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SSC Pure Mathematics QUESTION #4088
Question 1
If \(\varphi: \mathbb{Z}_6 \to \mathbb{Z}_6\) is a one-to-one (injective) homomorphism, then \(\ker(\varphi)\) is:
  • All of \(\mathbb{Z}_6\)
  • \(\{0, 2, 4\}\)
  • \(\{0, 3\}\)
  • \(\{0\}\)✔️
Correct Answer Explanation
A group homomorphism is injective if and only if its kernel is trivial (contains only the identity). Since \(\varphi\) is one-to-one, \(\ker(\varphi) = \{0\}\), the trivial subgroup of \(\mathbb{Z}_6\).