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The generalized dihedral group for an abelian group
Definition
Let be an abelian group, and let the cyclic group (Every cyclic group is isomorphic to or to for its finite order ) act on by inversion, . The generalized dihedral group of is the external semidirect product ( The external semidirect product )
The inversion map is an automorphism precisely because is abelian.
Depends on
Used by
- Dih(Cₙ)=Cₙ⋊ C₂ with inversion action has order 2n and the dihedral relations Corollary
- An extraspecial group of order 32 decomposes both as two quaternion factors and as two dihedral factors Counterexample
- Dih(C₂× C₂) is the direct product (C₂× C₂)× C₂ Example
- The commutator pairings of Dih(C₄) and Q₈ are the same, while the groups are not isomorphic Example
- The dihedral group of order eight has Cayley graphs that are a cycle of length eight and a cube Example
- The Frattini subgroups of the dihedral and quaternion groups of order eight Example
- The infinite dihedral group is quasi-isometric to ℤ, and to ℤ×ℤ/2 Example
- Dih(C₄) and Q₈ are extraspecial of order 8, with six and two solutions of x²=1 respectively Proposition
Dependency tree · two levels
12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. S. Milne, Group Theory (standard reference, not scraped)