The kernel and image of a group homomorphism

Depends on

• Monoid homomorphism and group homomorphism
• Injection, surjection, bijection

Used by

• A subgroup is normal if and only if it is the kernel of a group homomorphism Corollary
• 1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group Counterexample
• Group extensions, sections, complements, and split extensions Definition
• The alternating group Aₙ=ker(sgn) of even permutations Definition
• For n≥2, reduction ℤ→ℤ/n has kernel nℤ and realises ℤ/n by the first isomorphism theorem Example
• The trivial homomorphism G→ H has kernel G and image {e_H} Example
• The kernel of a finite direct sum is the intersection of the kernels Lemma
• Two elements have the same image under a homomorphism if and only if they lie in the same coset of its kernel Lemma
• A representation with kernel containing a normal subgroup factors through the quotient, and irreducibility is unchanged by inflation Proposition
• On the units, the Legendre symbol is the unique nontrivial homomorphism to {±1} Proposition
• The canonical projection R→ R/I is a surjective ring homomorphism with kernel I Proposition
• The two canonical maps into a central product are injective homomorphisms whose images commute, generate it, and meet in the identified centre Proposition
• A group homomorphism is injective if and only if its kernel is trivial Theorem
• 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
• Every group admits a presentation Theorem
• Homomorphisms out of a central product Theorem
• Internal central products are the images of external ones Theorem
• Left multiplication on G/H is transitive, has stabiliser H at H, and has kernel Core_G(H) Theorem
• The image of a group homomorphism is a subgroup and its kernel is a normal subgroup Theorem
• The kernel of a complex character agrees with the kernel of any representation affording it Theorem
• The kernel of a ring homomorphism is a two-sided ideal Theorem
• The kernel of the Jacobi map and the subgroup of unit squares Theorem
• The map g↦(x↦ gxg⁻¹) is a homomorphism G toAut(G) with kernel Z(G) and image Inn(G) Theorem
• Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group Theorem

Dependency tree · two levels

8 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, Homomorphisms (standard reference, not scraped)