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The image of a group homomorphism is a subgroup and its kernel is a normal subgroup
Statement
The image of a group homomorphism is a subgroup and its kernel is a normal subgroup.
For every group homomorphism , one has and .
Facts & Assumptions
Given: A group homomorphism .
The kernel is the inverse image of and the image is the set of values of (The kernel and image of a group homomorphism).
A group homomorphism preserves products and inverses (A group homomorphism automatically satisfies and , and for every ; for monoid homomorphisms preservation of the identity must be assumed).
A nonempty subset closed under is a subgroup (One-step subgroup test: a nonempty is a subgroup iff for all ; the identity and the inverses of are then those of ).
A subgroup is normal when for every (Normal subgroup: invariance under conjugation).
Proof
The image contains and, for , contains ; thus [L3] gives .
The kernel is a subgroup by the same calculation, and for one has , so ; applying this to gives equality.
The conjugation calculation in step 2.1 completes both assertions.
Depends on
- The kernel and image of a group homomorphism
- A group homomorphism automatically satisfies $f(e) = e'$ and $f(g^{-1}) = f(g)^{-1}$, and $f(g^{n}) = f(g)^{n}$ for every $n \in \mathbb{Z}$; for monoid homomorphisms preservation of the identity must be assumed
- One-step subgroup test: a nonempty $H \subseteq G$ is a subgroup iff $gh^{-1} \in H$ for all $g, h \in H$; the identity and the inverses of $H$ are then those of $G$
- Normal subgroup: invariance under conjugation
Used by
- A subgroup is normal if and only if it is the kernel of a group homomorphism Corollary
- Aₙ is normal in Sₙ; for n≥2, 2 |Aₙ|=n!, while Aₙ=Sₙ for n=0,1 Corollary
- There are exactly two isomorphism classes of groups of order 105 Corollary
- 1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group Counterexample
- The alternating group Aₙ=ker(sgn) of even permutations Definition
- For n≥2, Sₙ≅ Aₙ⋊ C₂ using any transposition complement Example
- The canonical group solution set on a two-element set Example
- The two-circle wedge has both regular and nonregular connected three-sheeted coverings Example
- If p<q are primes and |G|=pq, then G has a normal subgroup of order q Lemma
- In a group extension the kernel is normal and the quotient recovers the base Lemma
- Cayley's theorem: every group G is isomorphic to a subgroup of Sym(G) Theorem
- Correspondence theorem: subgroups of G/N correspond to subgroups of G containing N, with normality preserved Theorem
- Every group admits a presentation Theorem
- Every group of order 105 has normal Sylow 5- and 7-subgroups and is not simple Theorem
- Every group of order 30 has normal Sylow 3- and 5-subgroups and is not simple Theorem
- First isomorphism theorem for groups: G/ker f congimf Theorem
- Möbius transformations form a group and identify with the projective linear quotient of GL₂(C) Theorem
- Normal-subgroup quotients of a fixed free group give a canonical solution set for the underlying-set functor on groups Theorem
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group Theorem
Dependency tree · two levels
22 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)