The external semidirect product $N{⋊}_{\alpha }H$

Depends on

• An action of a group $H$ on a group $N$ by automorphisms
• Group and abelian group

Used by

• Mackey little-group reduction for an abelian normal subgroup Corollary
• The unitary dual need not be Hausdorff Counterexample
• Affine root hyperplanes, coroot translations, alcoves, and the affine reflection group Definition
• Affine, almost simple, diagonal, product action, and twisted wreath types Definition
• Connected Poincaré covers and positive translation energy Definition
• p-elementary and p-hyperelementary finite groups Definition
• The generalized dihedral group Dih(A)=A⋊ C₂ for an abelian group A Definition
• The graph subgroup attached to a map into a semidirect product Definition
• The holomorph Hol(G)=G rtimesAut(G) Definition
• The imprimitive wreath product of permutation groups Definition
• The modular group of order p³ as a semidirect product C_p²⋊ Cₚ Definition
• Kernel-conjugate complements differ by a principal crossed homomorphism Example
• Little groups for the real ax+b group and its orientation-preserving subgroup Example
• Small elementary and hyperelementary groups Example
• Sylow subgroups of Aff(ℤ/5): n₂=5 and n₅=1 Example
• The extended affine Weyl group and non-trivial alcove stabilizers Example
• The finite Heisenberg group is the unique Sylow p-subgroup of its coordinate upper-triangular group Example
• The four isomorphism types of groups of order 30 Example
• The unique Sylow p-subgroup of Aff(ℤ/p²) Example
• Under the ultrafilter lemma, the standard lamplighter group is amenable Example
• False statement: every group of order 42 has a normal Sylow 2-subgroup False statement
• Actions changed by automorphisms of the kernel and complement give isomorphic semidirect products Lemma
• Affine reflections: translation form, involutivity, local finiteness, and Wₐ=Q^∨⋊ W Lemma
• Fourier sample from a generalized-dihedral reflection coset state Lemma
• Heisenberg group polarization and the fixed-central-character orbit Lemma
• Isaacs' linear-character step Lemma
• Mackey induction classifies regular Euclidean dual actions Lemma
• Relative property (T) for SL2(R) semidirect R2 Lemma
• Hidden shift as generalized-dihedral hidden subgroup Theorem
• SLn(R) has property (T) for n at least three Theorem
• The semidirect-product multiplication makes N× H a group Theorem

Dependency tree · two levels

10 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

• Keith Conrad, Semidirect Products (standard reference, not scraped)