then the bijection is an automorphism (q.v.).

Intuitively, group theorists view two isomorphic groups as follows: For every element of a group there exists an element of such that "behaves in the same way" as (operates with other elements of the group in the same way as ). For instance, if generates then so does This implies, in particular, that and are in bijective correspondence. Thus, the definition of an isomorphism is quite natural.