Alphabeta MathSemidirect Products, Automorphism Groups and Split Extensions
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✓ 18 results · all verified · 11 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 7 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
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Definition
An action of a group H on a group N by automorphisms
Definition
The external semidirect product N⋊_α H
Theorem
The semidirect-product multiplication makes N× H a group
Proposition
The canonical copy of N is normal, the canonical copy of H is a complement, and conjugation induces the action
Proposition
The canonical semidirect decomposition is an internal direct product if and only if the defining action is trivial
Definition
An internal semidirect product and a complement to a normal subgroup
Theorem
Recognition theorem: G=NH with N is normal in G, N∩ H=1 exactly realises an external semidirect product
Definition
Group extensions, sections, complements, and split extensions
Theorem
Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent
Definition
The quaternion group Q₈={±1,± i,± j,± k} inside the nonzero quaternions
Proposition
Q₈ is a subgroup of ℍ^× with eight elements, and -1 is its only element of order 2
Definition
The outer automorphism group Out(G)=Aut(G)/Inn(G)
Lemma
If K is characteristic in N and N is normal in G, then K is normal in G
Definition
The holomorph Hol(G)=G rtimesAut(G)
Proposition
The holomorph acts faithfully on G by affine permutations x↦ gα(x)
Proposition
A permutation group with a regular normal subgroup G embeds in Hol(G)
Theorem
Aut(Cₙ)≅(ℤ/nℤ)^×
Theorem
Aut(ℤⁿ)≅ GLₙ(ℤ) for every finite rank n
Lemma
Actions changed by automorphisms of the kernel and complement give isomorphic semidirect products
Definition
The generalized dihedral group Dih(A)=A⋊ C₂ for an abelian group A
Corollary
Dih(Cₙ)=Cₙ⋊ C₂ with inversion action has order 2n and the dihedral relations
Lemma
For prime q and d≥1, the congruence xᵈ≡1 (mod q) has at most d residue-class solutions
Lemma
If p<q are primes and |G|=pq, then G has a normal subgroup of order q
Lemma
For primes p<q, nontrivial actions of Cₚ on C_q exist exactly when p∣(q-1) and are unique up to automorphisms
Theorem
Classification of groups of order pq for primes p<q
Sources
- Keith Conrad, Semidirect Products
- Keith Conrad, Semidirect Products, Definition 3.1 and Theorem 4.1
- J. S. Milne, Group Theory
- J. S. Milne, Group Theory, Example 3.9(c)
- Peter J. Cameron, The Holomorph of a Group
- Martin R. Bridson and Karen Vogtmann, Automorphism groups of free groups, surface groups and free abelian groups
- Peter Hackman, Elementary Number Theory, Chapter C