In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, a function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the codomain, not to be confused with the range). ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

Definition

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 In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element...

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

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 In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...

f(x) = exp(x)

(see exponential function). The exponential function is one of the most important functions in mathematics. ...

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 Jump to: navigation, search Integers are whole numbers and thier opposites. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ...

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). In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V), named after Felix Klein, is the group C2 × C2, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...

The group (R,+) is isomorphic to the group (C,+) of all complex numbers with addition. The group (C^*,·) of non-zero complex numbers with multiplication as operation is isomorphic to the group S¹ mentioned above. For these last two examples, one cannot construct concrete isomorphisms; the proofs rely on the axiom of choice. In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

Consequences

From the definition, it follows that any isomorphism f : G -> H 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. In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...

Automorphisms

An isomorphism from a group G to G itself is called an automorphism of G. The composition of two automorphisms 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. In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

An example in an Abelian group is replacing the group elements by their inverses. However, in groups where all elements are equal to their inverse this is the trivial automorphism, e.g. in the Klein four-group. For that group all permutations of the three non-identity elements are automorphisms. In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V), named after Felix Klein, is the group C2 × C2, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). ...

In C_p for a prime number p, one non-identity element can be replaced by any other, with corresponding changes in the other elements.

For non-Abelian groups there is for every group element an inner automorphism. In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1 for all x in G; where the conjugation is often denoted exponentially by ax. ...