An inner automorphism of a group 
is an automorphism
of the form 
,
where 
is a fixed element of 
. An outer automorphism of 
is an automorphism which cannot be expressed in this form
for 
,
but can be so expressed if 
belongs to a larger group containing 
.
For example, the automorphism of the symmetric group 
which maps the permutation 
to 
is an inner automorphism,
since 
.
However, it is an outer automorphism of the alternating
group 
since 
does not belong to 
and there is no element 
of 
such that 
.
