Consequences
From the definition, it follows that any isomorphism will map the identity element of G to the identity element of H,
that it will map inverses to inverses,
and more generally, nth powers to nth powers,
for all u in G, and that the inverse map is also a group isomorphism.
The relation "being isomorphic" satisfies all the axioms of an equivalence relation. If f is an isomorphism between two groups G and H, then everything that is true about G that is only related to the group structure can be translated via f into a true ditto statement about H, and vice versa.
Read more about this topic: Group Isomorphism
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