Question

what is an automorphism? what is an automorphism? @Mathematics

Answer 1

In general, a homomorphism is a map between two examples of the same sort of algebraic structure.

For example, if V and W are vector spaces. Then a linear map L : V --> W is a vector space homomorphism. The map L "commutes" with the vector space properties and that is what defines it as a vector space homomorphism.

If G and H are groups, a group homomorphism f : G --> H is a function G --> H that preserves the group properties.

An AUTOMORPHISM is a homomorphism where the domain and co-domain are equal.

T : V --> V
f : G --> G

Answer 2

in the context of groups it is an isomophism from the group to itself

Answer 3

what jamesj said

Answer 4

For example, the linear map R^3 --> R^3 given by
i -> j
j -> k
k -> i

is an example of an automorphism on R^3.

Answer 5

Thank you so much. That is all very helpful!

Answer 6

I should have also said that an automorphism is what sat73 mentioned: the function also an isomorphism. It's not sufficient that it be just an homomorphism with the same domain and co-domain, but also be 1:1 and onto.