Dihedral implies solvable

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., dihedral group) must also satisfy the second group property (i.e., solvable group)
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Statement

Dihedral groups are solvable groups.

Proof

The dihedral group of order $2n$ (or $\infty$) has a cyclic normal subgroup of order $n$ (or $\infty$), with abelian quotient group (indeed, isomorphic to cyclic group:Z2).

This cyclic group is solvable, so its normal series with abelian quotients proving normality may be appended with the dihedral group in order to create such a series for this group.

Example

The series $\{e\}\trianglelefteq C_{2}\trianglelefteq C_{4}\trianglelefteq C_{8}\trianglelefteq D_{16}$ shows that dihedral group:D16 is solvable.