Definition

A (group) isomorphism φ:G→G′\varphi: G \to G' is a bijective group homomorphism i.e. bijective map s.t. φ(a,b)=φ(a)φ(b)\varphi(a,b) = \varphi(a) \varphi(b) for all a,b∈Ga,b\in G.

Lemma

If φ:G→G′\varphi : G \to G' is an isomorphism, then the inverse map φ−1:G′→G\varphi^{-1}: G' \to G is also an isomorphism.

Definition

Two groups GG and G′G' are said to be isomorphic if there exists an isomorphism φ\varphi between the two groups; this may be indicated as G≈G′G \approx G'.

References

  1. M. Artin, Algebra, 2. ed. Boston, Mass. Munich: Pearson Education, Prentice Hall, 2011, p. 51.