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 onetoone 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:
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
In particular, for groups, an inner automorphism is an automorphism f_{g} : G > G given by a conjugacy by a fixed element g of the group G, that is, for all h in G, the map f_{g} is of the form f_{g}(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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