A group automorphism is an isomorphism from a group to itself. If 
is a finite multiplicative
group, an automorphism of 
can be described as a way of rewriting its multiplication
table without altering its pattern of repeated elements. For example, the multiplication table of the group of 4th roots of unity 
can be written as shown above, which means that
the map defined by
 |
is an automorphism of 
.
The map 
is also a group automorphism for 
as long as 
is not congruent to 0. Conjugating by a fixed element 
is a group automorphism called an inner automorphism.
In general, the automorphism group of an algebraic object 
, like a ring or field,
is the set of isomorphisms of that object 
, and is denoted 
. It forms a group by composition of maps. For a fixed
group 
,
the collection of group automorphisms is the automorphism group 
.
