group of the octonions (O) is the exceptional Lie group G2. In graph theory an automorphism of a graph is a permutation of the nodes that preserves edges and non-edges. In particular, if two nodes are joined by an edge, so are their images under the permutation. In geometry, an automorphism may be called a motion of the space. Specialized terminology is also used: In metric geometry an automorphism is a self-isometry. The automorphism group is also called the isometry group. In the category of Riemann surfaces, an automorphism is a biholomorphic map (also called a conformal map), from a surface to itself. For example, the automorphisms of the Riemann sphere are Möbius transformations. An automorphism of a differentiable manifold M is a diffeomorphism from M to itself. The automorphism group is sometimes denoted Diff(M). In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism (see homeomorphism group). In this example it is not sufficient for a morphism to be bijective to be an isomorphism. History One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus, where he discovered an order two automorphism, writing: so that is a new fifth root of unity, connected with the former fifth root by relations of perfect reciprocity. Inner and outer automorphisms In some categories—notably groups, rings, and 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. For each element a of a group G, conjugation by a is the operation given by (or a−1ga; usage varies). One can easily check that conjugation by a is a group automorphism. The inner automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma. The other automorphisms are called outer automorphisms. The quotient group is usually denoted by Out(G); the non-trivial elements are the cosets that contain the outer automorphisms. The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different. See also Antiautomorphism Automorphism (in Sudoku puzzles) Characteristic subgroup Endomorphism ring Frobenius automorphism Morphism