A regular triangle looks more symmetric than a nonequilateral triangle in the plane. The notion of symmetry can be attached to any mathematical object or set with some additional structure. This structure need not necessarily be algebraic, but can also be, for example, a graph. An isomorphism mapping the structure into itself is called an automorphism. The set of all automorphisms of a structure is a group with respect to composition of maps. This group represents the symmetry of the structure. In this section we will study automorphism groups of some structures. Such symmetry groups are important for determining and investigating regular structures in nature, like molecules and crystals.
We recall that a graph consists of a vertex set V and an edge set E, whose elements are subsets of V of size 2.
-
An automorphism of a
graph
(V,E)
is a bijective map V -> V
satisfying
if {v, w} 
E then
{g(v),
g(w)} E.
- Let K be a ring, field, group or monoid. An automorphism of K is an isomorphism K -> K.
The automorphisms of a given structure form a group.