Alphabeta MathSemidirect Products, Automorphism Groups and Split Extensions — Examples
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How 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.
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✓ 16 results · all verified · 13 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 3 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Free Groups and Presentations
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Semidirect Products, Automorphism Groups and Split Extensions
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Example
S₃≅ C₃⋊ C₂ via inversion
Example
The affine group of the real line is ℝ⋊ℝ^×
Example
For n≥2, Sₙ≅ Aₙ⋊ C₂ using any transposition complement
Example
Dih(C₂× C₂) is the direct product (C₂× C₂)× C₂
Example
Hol(C₈) is the group of affine maps x↦ ax+b with a∈{1,3,5,7}
Example
Aut(C₈)≅ C₂× C₂
Example
A shear and a quarter-turn exhibit noncommuting automorphisms of ℤ²
Example
Hol(C₂× C₂)≅ S₄
Example
There is a unique nonabelian group of order 21, namely C₇⋊ C₃ with multiplication by 2
Counterexample
1→ Cₚ→ C_p²→ Cₚ→1 does not split
Counterexample
1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group
Counterexample
The reflection complement in C₃⋊ C₂≅ S₃ is not normal
Counterexample
A subgroup of an abelian group need not be characteristic
False statement
False: every short exact sequence of groups splits
False statement
False: the kernel and quotient determine a group extension up to isomorphism
False statement