Endgroep

Endgroep is a term used in Dutch mathematics to denote the endomorphism group of a given object A. It consists of all endomorphisms, i.e., structure-preserving maps from A to itself, with the operation of composition as the binary operation. The exact algebraic structure of End(A) depends on what A is.

For a group G, End(G) is the set of all group homomorphisms from G to G. Under composition, End(G) forms a monoid; it need not be a group because endomorphisms are not always invertible. The subset Aut(G) of automorphisms—the invertible endomorphisms—forms a group. If A carries extra structure, such as being an abelian group or a module over a ring, End(A) often carries a richer algebraic structure, typically a ring under pointwise addition and composition. If A is a vector space over a field F, End_F(A) is an F-algebra and is isomorphic to a full matrix algebra in finite dimensions.

Common examples illustrate these ideas. The endomorphisms of a finite cyclic group Z/nZ correspond to multiplication by an integer mod n, so End(Z/nZ) ≅ Z/nZ as rings, while Aut(Z/nZ) ≅ (Z/nZ)×. For a finite-dimensional vector space V of dimension n over F, End_F(V) ≅ M_n(F). These endomorphism structures appear throughout algebra and representation theory, encoding how an object maps to itself while preserving its essential structure.

See also Aut(G), End(A) in category theory, and Endomorphism ring. The Dutch term endgroep is commonly used to refer to End(A) in specific contexts.