An isomorphism is a morphism with a two-sided inverse that is also a morphism. Two objects are isomorphic when such a map exists between them. For groups, rings, and vector spaces, a bijective structure-preserving map has a structure-preserving inverse; in other settings the inverse condition needs separate attention, as for a homeomorphism between topological spaces.
For example, is a group isomorphism from the additive integers to the additive even integers. It is bijective onto that target, satisfies , and has inverse . The two groups have the same addition pattern even though they are different sets. It is not a ring isomorphism for their inherited multiplications, since while .
A group isomorphism is a bijective group homomorphism . It matches the entire group operation: , with each product taken in its own group. Its inverse is automatically a group homomorphism. A group automorphism is the special case .[ref]
The target is part of the claim. The doubling example under isomorphism is a group isomorphism , between the additive integers and the additive even integers. Exactly the same rule , declared as , is an injective endomorphism that misses and every other odd integer, so is not a group automorphism. Changing the declared target changes surjectivity, even though it does not change the formula or its compatibility with addition.
An isomorphism lets a calculation in one model solve the same problem in another. The example under group identifies addition modulo four with function composition of rotations of a square via . Through this group isomorphism, adding exponents corresponds exactly to composing motions. More generally, a group isomorphism preserves order of a group element, carries subgroups to subgroups, and carries a generating set of a group to a generating set of a group. Having equally many elements is insufficient: the permutation in group automorphism already fails compatibility on two inputs.
Why does the inverse preserve the group operation? Write and , possible because of surjectivity. ThenThis proves the inverse condition directly. Analogous calculations work for each operation of a ring and for addition and scalar multiplication in a vector space. The argument relies on preserving those operations; it supplies no general continuity claim for the inverse of a continuous bijection between topological spaces.
A ring isomorphism is a bijective ring homomorphism. It preserves all ring operations and has a ring-homomorphic inverse.
Articles by others on the same topic
Isomorphism is a concept that appears in various fields such as mathematics, computer science, and social science, and it generally refers to a kind of equivalence or similarity in structure between two entities. Here are a few specific contexts in which the term is often used: 1. **Mathematics**: In mathematics, particularly in algebra and topology, an isomorphism is a mapping between two structures that preserves the operations and relations of the structures.
Something analogous to a group isomorphism, but that preserves whatever properties the given algebraic object has. E.g. for a field, we also have to preserve multiplication in addition to addition.
Other common examples include isomorphisms of vector spaces and field. But since both of those two are much simpler than groups in classification, as they are both determined by number of elements/dimension alone, see:
we tend to not talk about isomorphisms so much in those contexts.