GATE CS 2021 Set 1 Q34 — Monoids and Groups

To determine the cyclicity of the group $G$ and its subgroup $H$, we analyze their properties based on group theory principles:

1. Cyclicity of $G$:

   The order of the group is $|G| = 6$. Up to isomorphism, there are only two groups of order $6$:

   • The cyclic group $\mathbb{Z}_6$ (which is cyclic).
   • The symmetric group $S_3$ (which is non-abelian and therefore not cyclic).

   Since $G$ can be isomorphic to $S_3$, $G$ may not be cyclic.

2. Cyclicity of $H$:

   By Lagrange's theorem, the order of any subgroup $H$ must divide the order of the group $G$. Thus, $|H|$ must be a divisor of $6$.

   We are given that $1 < |H| < 6$. The only divisors of $6$ in this range are $2$ and $3$.

   Since both $2$ and $3$ are prime numbers, and any group of prime order is always cyclic, $H$ must be cyclic.

Therefore, $G$ may not be cyclic, but $H$ is always cyclic.