Math 2.0

Group theory and generalizations Definition

Image of a homomorphism

Builds on Group homomorphismSubgroup

Definition

Let GG and HH be groups and let φ ⁣:G→H\varphi \colon G \to H be a group homomorphism. The image of φ\varphi is the set im⁡φ=φ(G)={φ(g):g∈G}\operatorname{im} \varphi = \varphi(G) = \{\varphi(g) : g \in G\}.

Remarks

The image is a subgroup of HH, because φ(a)φ(b)−1=φ(ab−1)\varphi(a) \varphi(b)^{-1} = \varphi(ab^{-1}). The homomorphism φ\varphi is surjective exactly when im⁡φ=H\operatorname{im} \varphi = H. The image of the map n↦2nn \mapsto 2^n from ℤ\mathbb{Z} to ℚ×\mathbb{Q}^\times is the set of integer powers of 22.

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8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Group_homomorphism, Image and kernel
About this page
Tag0112DTS464
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Kinddefinition
WritingW0 stub
MSC20A05
DefinesImage of a homomorphism
UsesGroup homomorphism, Subgroup

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