Generators of Cyclic Groups
Cyclic groups usually have more than one generator – at least if the group has more than one member.
The set
is cyclic group under addition mod 5, generated by the element 1 but also by the elements 2, 3 and 4.
The set
is a cyclic group under addition mod 6, generated by the element 1 but also by the elements 5.
Every cyclic group of order
is isomorphic to the addition group
The following theorem gives the condition under which an element of a cyclic group may generate the group.
Let
be a cyclic group of order
Then
if and only if the greatest common denominator of
andis 1 (
).
Proof: Ifwe can write
for integers
and
Then
Then
so all powers of
belong to
henceand
generates
Suppose then that
Write
and
so that the order of
showing thatdoes not generate