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What is an isomorphism?

What is an isomorphism? Here are some definitions.

Noun
  1. (group algebra) A bijection f such that both f and its inverse f −1 are homomorphisms, that is, structure-preserving mappings.
  2. (computer science) a one-to-one correspondence between all the elements of two sets, e.g. the instances of two classes, or the records in two datasets
  3. (category theory) A morphism which has an inverse; the composition of the morphism and its inverse yields either one of two identity morphisms (depending on the order of composition).
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Examples
The dog says that there are no surfaces, just occasional strata of unusual density, and that there can't be abstaction because deep isomorphism is always required.
In this case H would contain a self-conjugate subgroup, and the isomorphism is multiple.
If properly presented, an analogy may hide an isomorphism, and so may satisfy the three preceding criteria.
Indeed, in mathematics, isomorphism is a perfect one-to-one, bijective mapping between two structures or sets.
Homomorphisms, quotient groups, isomorphism theorems and permutation groups.
There must be a determinate and expressible structural isomorphism, even though one could not say that the blueprint realizes the form of the house.

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