Group isomorphism

In abstract algebra, given two 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.

Table of contents

1 Examples 2 Consequences 3 Automorphisms

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.