Editing Group isomorphism (section)

== Consequences ==

From the definition, it follows that any isomorphism <math>f : G \to H</math> will map the identity element of <math>G</math> to the identity element of <math>H,</math> 
<math display="block">f(e_G) = e_H,</math>
that it will map [[inverse element|inverses]] to inverses,
<math display="block">f(u^{-1}) = f(u)^{-1} \quad \text{ for all } u \in G,</math>
and more generally, <math>n</math>th powers to <math>n</math>th powers,
<math display="block">f(u^n)= f(u)^n \quad \text{ for all } u \in G,</math> 
and that the inverse map <math>f^{-1} : H \to G</math> is also a group isomorphism.

The [[relation (mathematics)|relation]] "being isomorphic" is an [[equivalence relation]]. If <math>f</math> is an isomorphism between two groups <math>G</math> and <math>H,</math> then everything that is true about <math>G</math> that is only related to the group structure can be translated via <math>f</math> into a true ditto statement about <math>H,</math> and vice versa.