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SSC Pure Mathematics QUESTION #11202
Question 1
Let $G = \langle a: a^{12} = e\rangle$. Then $G =$
  • $\langle a^5\rangle$✔️
  • $\langle a^6\rangle$
  • $\langle a^2\rangle$
  • $\langle a^8\rangle$
Correct Answer Explanation
$G = \mathbb{Z}_{12}$ is cyclic of order 12. A generator $a^k$ generates $G$ iff $\gcd(k,12)=1$. Check: $\gcd(5,12)=1$ ✓; $\gcd(6,12)=6$ ✗; $\gcd(2,12)=2$ ✗; $\gcd(8,12)=4$ ✗. Therefore $G = \langle a^5\rangle$.