The group of all real numbers with addition, (R,+), is isomorphic to the group of all positive real numbers with multiplication (R^{+},×) via the isomorphism
The group Z of integers (with addition) is a subgroup of R, and the factor group R/Z is isomorphic to the group S^{1} of complex numbers of absolute value 1 (with multiplication); an isomorphism is given by
The Klein fourgroup is isomorphic to the direct product of two copies of Z/2Z (see modular arithmetic).
From the definition, it follows that f will map the identity element of G to the identity element of H,
The relation "being isomorphic" satisfies all the axioms of an equivalence relation. If f is an isomorphism between G and H, then everything that is true about G can be translated via f into a true statement about H, and vice versa.
An isomorphism from a group G to G is called an automorphism of G. The composition of two automorphism is again an automorphism, and with this operation the set of all automorphisms of a group G, denoted by Aut(G), forms itself a group, the automorphism group of G.
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