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Give an example of the dihedral group of smallest order
Chapter 7, Problem 35E(choose chapter or problem)
Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to \(Z_{12}\) and a subgroup isomorphic to \(Z_{20}\). No need to prove anything, but explain your reasoning.
Questions & Answers
QUESTION:
Give an example of the dihedral group of smallest order that contains a subgroup isomorphic to \(Z_{12}\) and a subgroup isomorphic to \(Z_{20}\). No need to prove anything, but explain your reasoning.
ANSWER:Step 1 of 2
To give an example of the dihedral group of smallest order that contains a subgroup isomorphic to \(\mathbb{Z}_{12}\) and a subgroup isomorphic to \(\mathbb{Z}_{20}\).
We know that \(\mathbb{Z}_{12}\) and \(\mathbb{Z}_{20}\) are cyclic groups and every isomorphic image of a cyclic group is also cyclic.
Therefore, we consider the cyclic subgroup of rotations as it is known that any reflection has order 2.
Thus, the order of subgroup of rotations in \(D_n\) is \(n\).