# Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

### Other articles related to "automorphism, automorphisms":

Asymmetric Graph
... Formally, an automorphism of a graph is a permutation p of its vertices with the property that any two vertices u and v are adjacent if and only if p(u) and p(v) are adjacent ... The identity mapping of a graph onto itself is always an automorphism, and is called the trivial automorphism of the graph ... An asymmetric graph is a graph for which there are no other automorphisms ...
Inner and Outer Automorphisms
... Lie algebras—it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms ... In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself ... One can easily check that conjugation by a is a group automorphism ...
Nielsen Transformation - Applications - Automorphism Groups
... In (Nielsen 1924), it is shown that the automorphism defined by the elementary Nielsen transformations generate the full automorphism group of a finitely generated free group ... Neumann used these ideas to give finite presentations of the automorphism groups of free groups ... a finite group (not necessarily free), not every automorphism is given by a Nielsen transformation, but for every automorphism, there is a generating set where the automorphism is given by a Nielsen transformation ...
IA Automorphism
... In mathematics, in the realm of group theory, an IA automorphism of a group is an automorphism that acts as identity on the abelianization ... An IA automorphism is thus an automorphism that sends each coset of the commutator subgroup to itself ... The IA automorphisms of a group form a subgroup of the automorphism group ...
Automorphisms Of The Symmetric And Alternating Groups - Small n - Alternating
... For n = 1 and 2, A1 = A2 = 1 is trivial, so the automorphism group is also trivial ... A3 = C3 = Z/3 is abelian (and cyclic) the automorphism group is GL(1, Z/3*) = C2, and the inner automorphism group is trivial (because it is abelian) ...