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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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For , the cosets form a group with identity and inverse
Statement
Let . The left cosets form a group under
Its identity is , and the inverse of is .
Facts & Assumptions
Given: A group and a normal subgroup .
Coset multiplication is well defined when is normal (Coset multiplication is well defined if and only if is normal).
The quotient set consists of the left cosets of , with the proposed product (The quotient group and coset product ).
A group operation is associative, has a two-sided identity, and gives every element a two-sided inverse (Group and abelian group).
Proof
By [L1], the formula in [F1] is a binary operation on the coset set, independent of representatives.
For , one has . Also , so is the identity.
The products and both equal , so is the inverse of .
Steps 1.1 through 1.3 verify the binary operation, associativity, identity, and inverse axioms; therefore is a group with the stated identity and inverses.
Depends on
Used by
- Every quotient group of an abelian group is abelian Corollary
- If [G:N] is finite then |G/N|=[G:N]; for finite G this equals |G|/|N| Corollary
- 1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group Counterexample
- G/{e} reproduces G, while G/G is the one-element quotient group Example
- The three-cycle subgroup of Sym({1,2,3}) is normal and its quotient has two elements Example
- If K is normal in G, N is normal in G and K⊆ N, then N/K is normal in G/K Lemma
- A representation with kernel containing a normal subgroup factors through the quotient, and irreducibility is unchanged by inflation Proposition
- For every n∈ℕ, the congruence-class group (ℤ/n,+) is the quotient group (ℤ,+)/nℤ Proposition
- The canonical projection π:G→ G/N, π(g)=gN, is a surjective group homomorphism Proposition
- Φ(P/N)=Φ(P)N/N for a normal subgroup of a finite p-group Proposition
- For a two-sided ideal I, the additive cosets form a ring R/I with identity 1+I Theorem
- G/N is abelian if and only if [G,G]⊆ N Theorem
- The quotient action is well defined and makes M/N a module Theorem
Cited to discharge well-definedness by The quotient group G/N and coset product (gN)(hN)=ghN.
Dependency tree · two levels
14 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
- T. W. Judson, Abstract Algebra: Theory and Applications, Factor Groups and Normal Subgroups (standard reference, not scraped)