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The canonical projection , , is a surjective group homomorphism
Statement
Let . The canonical projection
is a surjective group homomorphism.
Facts & Assumptions
Given: A group , a normal subgroup , and the quotient group .
The quotient product is (For , the cosets form a group with identity and inverse ).
A group homomorphism satisfies for all (Monoid homomorphism and group homomorphism).
A function is surjective if every equals for some (Injection, surjection, bijection).
Every left coset of has the form for a representative (Left and right cosets and of a subgroup).
Proof
For , one has , so is a group homomorphism.
Every element of is a coset for some , so is surjective.
Hence the canonical projection is a surjective group homomorphism.
Depends on
Used by
- A subgroup is normal if and only if it is the kernel of a group homomorphism Corollary
- The abelianisation Gᵃᵇ:=G/[G,G] and its canonical map Definition
- A representation with kernel containing a normal subgroup factors through the quotient, and irreducibility is unchanged by inflation Proposition
- The canonical map M→ M/N is a surjective module homomorphism with kernel N; thus every submodule is a kernel Proposition
- The canonical projection R→ R/I is a surjective ring homomorphism with kernel I Proposition
- A homomorphism that kills a normal subgroup factors uniquely through the quotient group Theorem
- Correspondence theorem: subgroups of G/N correspond to subgroups of G containing N, with normality preserved Theorem
- If [G:H]=n<∞, then Core_G(H) is normal in G, [G:Core_G(H)]∣ n!, and only finitely many subgroups contain H Theorem
- Second isomorphism theorem for groups: H/(H∩ N)≅ HN/N Theorem
- Subgroups and quotients of solvable groups are solvable Theorem
- Third isomorphism theorem for groups: (G/K)/(N/K)≅ G/N Theorem
- Two finite presentations define isomorphic groups if and only if a finite sequence of Tietze transformations and inverses connects them Theorem
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group Theorem
Dependency tree · two levels
13 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
- Ernst, An Inquiry-Based Approach to Abstract Algebra, Quotients of Groups (standard reference, not scraped)