GATE CSE 2021 Set 1 Q44: Cyclicity of a group of order 6 and its proper subgroups

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?

Try it: multiple choice: exactly one correct option

Wrong answer costs 2/3 mark in GATE.

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Answer: B (official GATE 2021 final answer key)

Concept used: By Lagrange, $|H|$ is 2 or 3, a prime; every group of prime order is cyclic. $S_3$ has order 6 and is not cyclic.

First step: List the possible orders of $H$ by Lagrange's theorem, and recall a non-abelian group of order 6.

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Source: official GATE 2021 Set 1 CS question paper and final answer key, organised by IIT Bombay. Question paper · Answer key