Editing Automorphism (section)

==Definition==
In an [[algebraic structure]] such as a [[group (mathematics)|group]], a [[ring (mathematics)|ring]], or [[vector space]], an ''automorphism'' is simply a [[bijective]] [[homomorphism]] of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, [[group homomorphism]], [[ring homomorphism]], and [[linear operator]].) 

More generally, for an object in some [[category (mathematics)|category]], an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism <math>f: X\to X</math> is an automorphism if there is a morphism <math>g: X\to X</math> such that <math>g\circ f= f\circ g = \operatorname {id}_X,</math> where <math>\operatorname {id}_X</math> is the [[identity morphism]]  of {{mvar|X}}. For algebraic structures, the two definitions are equivalent; in this case, the identity morphism is simply the [[identity function]], and is often called the ''trivial automorphism''.