An invertible endomorphism of is called an automorphism. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. An isomorphism to itself is called an automorphism. Note that this particular automorphism (negative in abelian groups) agrees with complex conjugation. The identity morphism (identity mapping) is called the trivial automorphism in some contexts. Unlike ordinary rings, vertex algebras admit a notion of twisted module attached to an automorphism. The nontrivial outer automorphism in formula 9 interchanges these two classes and the corresponding icosahedra. The Galois group has two elements, the identity automorphism and the complex conjugation automorphism. Then it generates a subgroup of the automorphism group of . The Lie algebra of the automorphism groups can be written down immediately. PGL(3,4) has an outer automorphism induced by transposing conjugate elements in F (the field automorphism). Let be the automorphism which is given by the mapping acting on pure vectors. That is to say that there exists a color interchanging automorphism of this graph. L(3)&colon;2 Two classes, fused by an outer automorphism. The group contains only the identity automorphism.