Group isomorphism

Given two mathematical groups (G, *) and (H, @) a group isomorphism from (G, *) to (H, @) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function f : G -> H such that for all u and v in G it holds that

f(u * v) = f(u) @ f(v).

If there exists an isomorphism between the groups G and H, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

Examples

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

f(x) = exp(x)

(see exponential function).

The group Z of integers (with addition) is a subgroup of R, and the factor group R/Z is isomorphic to the group S¹ of complex numbers of absolute value 1 (with multiplication); an isomorphism is given by

f(x + Z) = exp(2πxi)

for every x in R.

The Klein four-group is isomorphic to the direct product of two copies of Z/2Z (see modular arithmetic).

Consequences

From the definition, it follows that f will map the identity element of G to the identity element of H,

f(e_G) = e_H

that it will map inverses to inverses,

f(u^-1) = f(u)^-1

for all u in G, and that the inverse map f^-1 : H -> G is also a group isomorphism.

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.

Automorphisms

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.