I just showed that $D_4$ is solvable since it is isomorphic to $\mathbb{Z}_2 x \mathbb{Z}_2$ so since $\mathbb{Z}_2$ is solvable so is $D_4$. Does something similar extend to $D_5$ or is there a better way of showing both?
2026-03-28 11:35:03.1774697703
Is the Dihedral Group $D_5$ solvable?
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All dihedral groups $D_n$ have an abelian normal subgroup order $n$ (in fact it is cyclic) with the corrresponding quotient abelian (of order 2, to be precise) and so they are solvable.