WOLFRAM|DEMONSTRATIONS PROJECT

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n
20
k in
α
3
3
The homomorphism α
3
: 
20
→ 
20
defined by α
3
(s) = 3s is
an automorphism of 
20
.
The kernel is {0} and the range is

20
.
The group

n
consists of the residue classes of the integers modulo
n
under addition. This Demonstration illustrates the action of a group homomorphism
α:

n


n
. Such a mapping must have the form
α(x)=ax
for some
a
in

n
. The homomorphism
α
is an automorphism if and only if
a=α(1)
is a generator of the cyclic group

n
.