Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Automorphism
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==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''.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)