Endomorphism of a group
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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 with symbols
Let be a group. A map from to itself is termed an endomorphism of if it satisfies all of the following conditions:
- whenever and are both in
Actually, the second and third condition follow from the first (refer equivalence of definitions of group homomorphism).
Relation with other properties
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.
The identity map is always an endomorphism.
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.