Math 2.0

Group theory and generalizations Example

A translation of the integers is not a homomorphism

Builds on IntegersGroup homomorphism

Example

Let G=(ℤ,+)G = (\mathbb{Z}, +) be the integers with addition. Define t:G→Gt : G \to G by t(u)=u+1t(u) = u + 1. A homomorphism sends the identity to the identity. But t(0)=1t(0) = 1, and 11 is not the identity 00. We can also compute directly: t(0+0)=1t(0 + 0) = 1, but t(0)+t(0)=2t(0) + t(0) = 2. Thus tt is not a homomorphism.

Remarks

tt is a bijection, so a bijection between groups need not be an isomorphism.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Group_homomorphism, Properties
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Not an instance ofGroup homomorphism (note: it sends the identity 0 to 1.)
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