Alphabeta Math

Normal Subgroups and Quotient Groups

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✓ 13 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Definition

Normal subgroup: invariance under conjugation

Theorem

Equivalent characterisations of a normal subgroup by conjugates and left and right cosets

Corollary

Every subgroup of an abelian group is normal

Theorem

Every subgroup of index two is normal

Lemma

The intersection of a nonempty family of normal subgroups is normal

Definition

The normal closure of a subset of a group

Definition

The center Z(G) of a group

Lemma

The center of a group is a normal subgroup

Definition

Commutators [g,h]=ghg⁻¹h⁻¹ and the commutator subgroup [G,G]

Lemma

The commutator subgroup is normal

Definition

The quotient group G/N and coset product (gN)(hN)=ghN

Theorem

Coset multiplication (gH)(hH)=ghH is well defined if and only if H is normal

Theorem

For N is normal in G, the cosets form a group with identity N and inverse (gN)⁻¹=g⁻¹N

Proposition

The canonical projection π:G→ G/N, π(g)=gN, is a surjective group homomorphism

Corollary

If [G:N] is finite then |G/N|=[G:N]; for finite G this equals |G|/|N|

Theorem

G/N is abelian if and only if [G,G]⊆ N

Corollary

Every quotient group of an abelian group is abelian

Proposition

For every n∈ℕ, the congruence-class group (ℤ/n,+) is the quotient group (ℤ,+)/nℤ

Sources