The Group Properties Wiki (pre-alpha)
TIP: Read more about how the definition in Groupprops is structured
ABOUT US: We use a Creative Commons license. All our content is free to reuse, with attribution. Learn more
ALSO CHECK OUT: Commalg: The Commutative Algebra Wiki
Endomorphism of a group
From Groupprops
|
This article is about a basic definition in group theory. The article text may, however, contain more material. Rate its utility as a basic definition article on the talk page
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in group theory | Go through a guided tour for beginners to this wiki
This article defines a function property, viz a property of functions from a group to itself
Endomorphism redirects here. For the more general notion, refer endomorphism of a universal algebra
Definition
Symbol-free definition
An endomorphism of a group is a homomorphism from the group to itself.
Definition with symbols
Let G be a group. A map σ from G to itself is termed an endomorphism of G if it satisfies all of the following conditions:
- σ(gh) = σ(g)σ(h) whenever g and h are both in G
- σ(e) = e
- σ(g − 1) = σ(g) − 1
Actually, the second and third condition follow from the first (refer equivalence of definitions of group homomorphism).
Relation with other properties
Weaker properties
Facts
Composition
The composite of two endomorphisms of a group is again an endomorphism of the group. This follwos from the fact that the composite of any two isomorphisms is an isomorphism.
Identity map
The identity map is always an endomorphism.
Monoid structure
Combining the fact that endomorphisms are closed under composition, and the fact that the identity map is an endomorphism, the endomorphisms of a group form a submonoid of the monoid of all functions from the group to itself. This submonoid is termed the endomorphism monoid. Its invertible elements are precisely automorphisms of the group.

