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Automorphism

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In mathematics, an automorphism of a mathematical object is a mapping which:

Very informally, an automorphism is a symmetry of the object, a way of showing its internal regularity (whichever side of a regular polygon you choose as it basis, it looks the same).

For example, in graph theory an automorphism of a graph is a permutation of the nodes that maps the graph to itself. In group theory, an automorphism of a group G is a bijective homomorphism of G onto itself (that is, a one-to-one map G -> G that preserves the group operation; informally, a way of shuffling the elements of the group which doesn't affect the structure).

The set of automorphisms of an object X together with the operation of function composition forms a group called the automorphism group of X, Aut(X). That this is indeed a group is simple to see:

  • Closure: composition of two bijections is a bijection, composition of homomorphisms is a homomorphism
  • Identity: the identity automorphism is simply "do nothing": the identity mapping of the object onto itself
  • Inverse: since an automorphism is by definition a bijection, it has an inverse. This satisfies the same properties, and is therefore an automorphism itself
  • Associativity: function composition is trivially associative

When it is possible to build transformation of an object by selecting one of its elements and applying operations to the object, one can separate

The latter being transformations for which there is no correspondence with an element of the object.

In particular, for groups, an inner automorphism is an automorphism fg : G -> G given by a conjugacy by a fixed element g of the group G, that is, for all h in G, the map fg is of the form fg(h) = g-1 hg. The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G). The quotient group Aut(G) / Inn(G) is usually denoted by Out(G).



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