An automorphism group is a mathematical concept in the field of abstract algebra. It consists of all the automorphisms of a given structure, such as a group, ring, or graph. An automorphism is a special type of isomorphism that maps a structure onto itself while preserving its operations and properties. The collection of these automorphisms forms a group under the operation of composition. The study of automorphism groups helps mathematicians understand the symmetries and structural properties of various mathematical objects. For example, in group theory, the automorphism group of a group G reveals how the elements of G can be rearranged without altering the group's structure. This concept is crucial in areas like algebraic topology and geometry.