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 semidirect-product multiplication makes a group
Statement
Let be an action by automorphisms. The multiplication
makes a group with identity and inverse
Facts & Assumptions
Given: Groups and a homomorphism .
The external semidirect-product multiplication is ( The external semidirect product ).
An action by automorphisms satisfies and , with every an automorphism of (An action of a group on a group by automorphisms).
A homomorphism preserves identities and inverses (A group homomorphism automatically satisfies and , and for every ; for monoid homomorphisms preservation of the identity must be assumed).
Proof
For three pairs, multiplication in either parenthesisation gives because is a homomorphism and . Thus the operation is associative.
Since is the identity and every preserves , the pair is a two-sided identity.
Put . Then and . Hence the displayed pair is the two-sided inverse.
Depends on
- The external semidirect product $N\rtimes_\alpha H$
- An action of a group $H$ on a group $N$ by automorphisms
- A group homomorphism automatically satisfies $f(e) = e'$ and $f(g^{-1}) = f(g)^{-1}$, and $f(g^{n}) = f(g)^{n}$ for every $n \in \mathbb{Z}$; for monoid homomorphisms preservation of the identity must be assumed
Used by
- Affine, almost simple, diagonal, product action, and twisted wreath types Definition
- The imprimitive wreath product of permutation groups Definition
- The modular group of order p³ as a semidirect product C_p²⋊ Cₚ Definition
- The affine group of the real line is ℝ⋊ℝ^× Example
- The finite Heisenberg group is the unique Sylow p-subgroup of its coordinate upper-triangular group Example
- A graph subgroup is a complement exactly for a crossed homomorphism Lemma
- Actions changed by automorphisms of the kernel and complement give isomorphic semidirect products Lemma
- Kernel conjugation by an element of the coefficient group corresponds to a principal crossed homomorphism Lemma
- The canonical copy of N is normal, the canonical copy of H is a complement, and conjugation induces the action Proposition
- Recognition theorem: G=NH with N is normal in G, N∩ H=1 exactly realises an external semidirect product Theorem
Cited to discharge well-definedness by The external semidirect product N⋊_α H.
Dependency tree · two levels
21 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)