eli5 what is an automorfism?
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It’s usually spelled automor**ph**ism in English.
An automorphism is an isomorphism from an object to itself. An isomorphism, in turn, is a homomorphism that has an inverse that is also a homomorphism. What a homomorphism is depends on what kind of object you’re studying, but roughly speaking, it’s a function that “preserves structure”. To give a few examples:
* A homomorphism of a set with some sort of operation on it preserves the operation, that is, f(x*y) = f(x)*f(y) (it doesn’t matter if you do the operation * before or after the function).
* A homomorphism of a metric space preserves distances, that is, the distance from x to y is the same as the distance from f(x) to f(y).
* A homomorphism of vector spaces preserves vector operations, that is, f(a**x** + b**y**) = a f(**x**) + b f(**y**). We usually just call this a linear function.
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As for why we care: automorphisms turn out to be really really important in algebra.
As an example, if you’re familiar with group theory, a group G’s automorphisms turn out to reveal a lot of information about the group itself, and the automorphisms actually have a relationship to G itself because each element *g* in G can be used to create a (not necessarily unique) automorphism of its own. Similarly, the automorphisms of a vector space correspond to all the n x n matrices.
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