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 
.
For a prime number 
and an integer 
, the map 
is also a group automorphism of 
whenever 
. 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 
.
