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Abstract Algebra

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Moderate

Group of Given Order: A Group of Order 18 Determined by Subgroups

ABSALG-1BSHON

Let $G$ be a group of order $18$ with $9$ subgroups of order $2$ and one cyclic subgroup of order $9$.

$G$ is isomorphic to which of the following?

A

A cyclic group.

B

$S_3\times (\mathbb{Z}/3\mathbb{Z})$

C

$D_9$ (dihedral group with $18$ elements)

D

$A_3\times (\mathbb{Z}/6\mathbb{Z})$