DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The generalized dihedral group $Dih (A) = A ⋊ C_{2}$ for an abelian group $A$

Definition

Let $A$ be an abelian group, and let the cyclic group $C_{2} = ⟨ s ⟩$ (Every cyclic group is isomorphic to $(Z, +)$ or to $(Z / n, +)$ for its finite order $n \geq 1$ ) act on $A$ by inversion, $s \cdot a = a^{- 1}$ . The generalized dihedral group of $A$ is the external semidirect product ( The external semidirect product $N ⋊_{α} H$ )

$Dih (A) = A ⋊ C_{2} .$

The inversion map is an automorphism precisely because $A$ is abelian.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 51 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Group Theory (standard reference, not scraped)

The generalized dihedral group Dih⁡(A)=A⋊C2Dih(A)=A⋊C2​ for an abelian group AA