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Concept
Group isomorphism
Summary
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.
Given two groups and a group isomorphism from to is a bijective group homomorphism from to Spelled out, this means that a group isomorphism is a bijective function such that for all and in it holds that
The two groups and are isomorphic if there exists an isomorphism from one to the other. This is written
Often shorter and simpler notations can be used. When the relevant group operations are understood, they are omitted and one writes
Sometimes one can even simply write 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.
Conversely, given a group a set and a bijection we can make a group by defining
If and then the bijection is an automorphism (q.v.).
Intuitively, group theorists view two isomorphic groups as follows: For every element of a group there exists an element of such that "behaves in the same way" as (operates with other elements of the group in the same way as ). For instance, if generates then so does This implies, in particular, that and 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).
In this section some notable examples of isomorphic groups are listed.
The group of all real numbers under addition, , is isomorphic to the group of positive real numbers under multiplication :
via the isomorphism .
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