Undergrad What is difference between transformations and automorphisms

[SVN]

Could you please help me to understand what is the difference between notions of «transformation» and «automorphism» (maybe it is more correct to talk about «inner automorphism»), if any? It looks like those two terms are used interchangeably.

By «transformation» I mean mapping from some set into itself (transformation groups in geometry), not notions like «Laplace transform(ation)» or «Legendre transform(ation)» where, as far as I know, the term is being used for historical reasons mainly.

 

[fresh_42]

Transformation normally refers to only linear mappings, aka functions. Of course this isn't a given as the term is not exactly defined out of context. Automorphisms on the other hand are precisely defined:

morphism - function which respects the structure: groups, rings, algebras, topologies, linearity etc. (in general: category)
endomorphism - morphism with the same set as domain and codomain
epimorphism - surjective morphism, surjection
monomorphism - injective morphism, injection
isomorphism - bijective morphism, bijection
automorphism - bijective endomorphism

Sometimes certain automorphisms have the additional property inner, which means they have a certain form themselves. Which, depends on the category.

 

[SVN]

@fresh_42 Will it be correct (albeit not quite rigourous, but for now I am trying to grasp the very idea) to say that transformations are automorphisms of space (set, manifold) arising in context of consideration of general linear group and subgroups thereof?

 

[fresh_42]

@fresh_42 Will it be correct (albeit not quite rigourous, but for now I am trying to grasp the very idea) to say that transformations are automorphisms of space (set, manifold) arising in context of consideration of general linear group and subgroups thereof?

As far as they are elements of a matrix group (matrix algebra), the term transformation is common.

If regular, then they are automatically automorphisms. But consider that a transformation in general doesn't have to be on the same vector space: a projection is also a (linear) transformation. So in a way - with respect to only linear functions - automorphisms are a subset of transformations. At last it will be necessary to consider if we speak about manifolds, where functions between different dimensions are not unusual. And don't forget, that by a transformation between manifolds it is very likely meant to refer to their linear tangent spaces, not the manifolds themselves. Also the word linear may often just be omitted, although there are also non-linear transformations thinkable, even if usually not called this way. All this shows, that the word transformation is rather context sensitive, but your are right, that in 99% of the cases it is simply a matrix (if given basis vectors). Transformation has the advantage, that we don't need to point out a basis.

If in doubt, i.e. no matrices around, it should be mentioned what is to be transformed.

 

[SVN]

@fresh_42 Pretty clear and exhaustively! Thank you very much!

 

[fresh_42]

Whole information is sound good for everyone who want to know about the different between them. But i like to add more content about both terms.

... which depends on the specific category in question. Group, ring, field, algebra, vector space, topological homomorphisms all have different properties. Homomorphism just says "same structure": different structure - different rules.