The GATE Grind

GATE 2021 CS – Question 44

Engineering Mathematics · Discrete Mathematics: Monoids and Groups · 2 marks · Multiple choice

Let $G$ be a group of order 6, and $H$ be a subgroup of $G$ such that $1 < |H| < 6$. Which one of the following options is correct?

  1. Both $G$ and $H$ are always cyclic.
  2. $G$ may not be cyclic, but $H$ is always cyclic.
  3. $G$ is always cyclic, but $H$ may not be cyclic.
  4. Both $G$ and $H$ may not be cyclic.

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Show answer and explanation

Correct answer: (B) $G$ may not be cyclic, but $H$ is always cyclic.

Explanation

A group of order 6 can be $S_3$, which is non-cyclic. By Lagrange, |H| is 2 or 3, which is prime, so H is always cyclic.