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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
- The alternating group Aₙ=ker(sgn) of even permutations Definition
- 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
- First isomorphism theorem for groups: G/ker fcongimf Theorem
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 48 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Ernst, An Inquiry-Based Approach to Abstract Algebra, Homomorphisms (standard reference, not scraped)