Automorphism

A automorphism of the group of four of Klein that is shown as a mapping between two Cayley graphics, a permutation in cycle notation and a mapping between two Cayley tables

In mathematics, an automorphism is an isomorphism of a mathematical object itself. Usually the set of automorphisms of an object can receive a group structure with the composition operation, such a group is called group of automorphisms and is, roughly speaking, the symmetry group of the object.

Examples

If the structures are sets, then the isomorphisms between two sets X, Y are simply bijective functions.

Automorphisms are bijective functions from X to X, that is, permutations of the set.

Considering the set Z of integers with the structure of abelian group (with the operation sum), automorphisms are bijective functions f:Z→Z such that f(x+and)=f(x)+f(and){displaystyle f(x+y)=f(x)+f(y)}. There are two unique functions with such property: f(x)=x{displaystyle f(x)=x} and f(x)=− − x{displaystyle f(x)=-x}.

If now we take the whole again Z integers but with the ring structure (summary operations and product) then the automorphisms will be bijective functions that fulfill f(x+and)=f(x)+f(and){displaystyle f(x+y)=f(x)+f(y)} and f(xand)=f(x)f(and){displaystyle f(xy)=f(x)f(y)}. In this case, the only possible function is identity, since f(x)=− − x{displaystyle f(x)=-x} only meets the first condition and not the second.

In all three cases, the automorphism group suggests some symmetry in the object. In the case of sets, since they lack structure, any rearrangement of their elements (permutations) is taken. In the case of integers, when only the structure of the addition is considered, a symmetry between positive and negative numbers is obtained, but such symmetry disappears when the structure imposed by multiplication is taken into account, since the behavior of the positive and negative numbers is different with respect to it.