Math 2.0

Group theory and generalizations Definition

Group isomorphism

Builds on Group homomorphism

Definition

Let GG and HH be groups. A group isomorphism from GG to HH is a group homomorphism φ ⁣:G→H\varphi \colon G \to H that is a bijection. The groups GG and HH are isomorphic, written G≅HG \cong H, when there is an isomorphism from GG to HH.

Remarks

The inverse map of an isomorphism is again a homomorphism, thus an isomorphism. Isomorphism is an equivalence relation on groups. The exponential map is an isomorphism from (ℝ,+)(\mathbb{R}, +) to the positive real numbers with multiplication. Isomorphic groups have the same order, and corresponding elements have the same order.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Group_isomorphism, Definition and notation
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Aliasesisomorphism-of-groups
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MSC20A05
DefinesGroup isomorphism
UsesGroup homomorphism

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