Definition:Homomorphism (Abstract Algebra)/Image
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Definition
Let $S$ and $T$ be algebraic structures.
Let $\phi: S \to T$ be a homomorphism from $S$ to $T$.
As a homomorphism is a mapping, the homomorphic image of $\phi$ is defined in the same way as the image of a mapping:
- $\Img \phi = \set {t \in T: \exists s \in S: t = \map \phi s}$
Linguistic Note
The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.
Thus homomorphism means similar structure.
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.4$. Kernel and image: $(2)$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 66$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Definition $8.12$