Alphabeta MathGroup Homomorphisms and the Isomorphism Theorems
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✓ 17 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Group isomorphisms, automorphisms and the set Aut(G)
Lemma
The inverse of a bijective group homomorphism is a group homomorphism
Definition
The kernel and image of a group homomorphism
Theorem
The image of a group homomorphism is a subgroup and its kernel is a normal subgroup
Theorem
A group homomorphism is injective if and only if its kernel is trivial
Corollary
A subgroup is normal if and only if it is the kernel of a group homomorphism
Lemma
Two elements have the same image under a homomorphism if and only if they lie in the same coset of its kernel
Theorem
A homomorphism that kills a normal subgroup factors uniquely through the quotient group
Theorem
First isomorphism theorem for groups: G/ker f congimf
Lemma
If H≤ G and N is normal in G, then HN is a subgroup and H∩ N is normal in H
Theorem
Second isomorphism theorem for groups: H/(H∩ N)≅ HN/N
Lemma
If K is normal in G, N is normal in G and K⊆ N, then N/K is normal in G/K
Theorem
Third isomorphism theorem for groups: (G/K)/(N/K)≅ G/N
Theorem
Correspondence theorem: subgroups of G/N correspond to subgroups of G containing N, with normality preserved
Theorem
The automorphisms of a group form a group under composition
Theorem
Conjugation x↦ gxg⁻¹ is an automorphism
Definition
Inner automorphisms and Inn(G)
Theorem
The map g↦(x↦ gxg⁻¹) is a homomorphism G toAut(G) with kernel Z(G) and image Inn(G)
Corollary
G/Z(G)congInn(G)
Theorem