Group isomorphism

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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.

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Definition and notation

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

The two groups (G, *) and (H, \odot) are isomorphic if an isomorphism exists. This is written:

Often shorter and more simple notations can be used. Often there is no ambiguity about the group operation, and it can be omitted:

Sometimes one can even simply write G = H. 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 (G, *), a set H, and a bijection f : G \rightarrow H, we can make H a group (H, \odot) by defining

If H = G and \odot = * then the bijection is an automorphism (q.v.)

Intuitively, group theorists view two isomorphic groups as follows: For every element g of a group G, there exists an element h of H such that h 'behaves in the same way' as g (operates with other elements of the group in the same way as g). For instance, if g generates G, then so does h. This implies in particular that G and H are in bijective correspondence. So the definition of an isomorphism is quite natural.

An isomorphism of groups may equivalently be defined as an invertible morphism in the category of groups, where invertible here means has a two-sided inverse.

Examples

  • The group of all real numbers with addition, (\mathbb{R},+), is isomorphic to the group of all positive real numbers with multiplication (\mathbb{R}+,×):

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