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.

Read more about Automorphism: Definition, Automorphism Group, Examples, History, Inner and Outer Automorphisms

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

Asymmetric Graph

... Formally, an

... 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

... Lie algebrasâ€”it is possible to separate

**Automorphism**s... Lie algebrasâ€”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 -

... In (Nielsen 1924), it is shown that the

**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

... In mathematics, in the realm of group theory, an 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 ...**Automorphism**s 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) ...

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