In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection 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.

Definition and notation

a group isomorphism is a bijective group homomorphism

are isomorphic if there exists an isomorphism from one to the other.

Whether such a notation is possible without confusion or ambiguity depends on context. For example, the equals sign is not very suitable when the groups are both subgroups of the same group. See also the examples.

and a bijection

then the bijection is an automorphism (q.v.).

generates are in bijective correspondence. Thus, the definition of an isomorphism is quite natural.

An isomorphism of groups may equivalently be defined as an invertible group homomorphism (the inverse function of a bijective group homomorphism is also a group homomorphism).

Examples

In this section some notable examples of isomorphic groups are listed.

• The group of all real numbers, is isomorphic to the group of positive real numbers:

  1. .

  of integers and the factor group of complex numbers of absolute value 1 (under multiplication):

• The Klein four-group is isomorphic to the direct product because it is a dihedral group.

• Generalizing this, for all odd

• is an infinite cyclic group is isomorphic to the integers (with the addition operation). From an algebraic point of view, this means that the set of all integers (with the addition operation) is the "only" infinite cyclic group.

Some groups can be proven to be isomorphic, relying on the axiom of choice, but the proof does not indicate how to construct a concrete isomorphism. Examples:

• of all complex numbers under addition.

• mentioned above.

Properties

The kernel is always {eG}, where eG is the identity

is abelian is abelian.

the order

is a locally finite group is locally finite.

The number of distinct groups (up to isomorphism) of order is given by sequence A000001 in the OEIS. The first few numbers are 0, 1, 1, 1 and 2 meaning that 4 is the lowest order with more than one group.

Cyclic groups

denotes addition modulo

Consequences

that it will map inverses is also a group isomorphism.

The relation "being isomorphic" is an equivalence relation and vice versa. == Automorphisms ==

to itself is called an automorphism

The image under an automorphism of a conjugacy class is always a conjugacy class (the same or another).

The composition itself forms a group, the automorphism group

For all abelian groups there is at least the automorphism that replaces the group elements by their inverses. However, in groups where all elements are equal to their inverses this is the trivial automorphism, e.g. in the Klein four-group. For that group all permutations).

for a prime number in that order or conversely.

so apart from the identity we can only interchange these.

automorphisms. They correspond to those of the Fano plane, of which the 7 points correspond to the 7 See also general linear group over finite fields.

For abelian groups, all non-trivial automorphisms are outer automorphisms.

Non-abelian groups have a non-trivial inner automorphism group, and possibly also outer automorphisms.

Automorphisms that send each element to within its own conjugacy class are called class-preserving. The set of all such automorphisms forms a normal subgroup of the automorphism group. All inner automorphisms are class-preserving.

See also

• Group isomorphism problem

References