Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The external semidirect product
Definition
Let and be groups (Group and abelian group), and let be an action by automorphisms (An action of a group on a group by automorphisms). The external semidirect product is the set with multiplication
When the action is clear, the subscript is omitted.
Depends on
Used by
- Affine, almost simple, diagonal, product action, and twisted wreath types 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
- Small elementary and hyperelementary groups Example
- Sylow subgroups of Aff(ℤ/5): n₂=5 and n₅=1 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
- Isaacs' linear-character step Lemma
- 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)