Cyclic Groups
A group G is called cyclic if there exists an element g in G such that
Since any group generated by an element in a group is a subgroup of that group, showing that the only subgroup of a group
that contains
isitself suffices to show thatis cyclic.
For example, if
is a group, then
and G is cyclic. In fact,is isomorphic to
with addition
For example,
corresponds to
We can use the isomorphism
defined by
For every positive integer
there is exactly one cyclic group (up to isomorphism) whose order is
and there is exactly one infinite cyclic group (the integers under addition). Hence, the cyclic groups are the simplest groups and they are completely classified.
Since the cyclic groups are abelian, they are often written additively and denoted
or
or C-n where n is the order, equal to the number of elements.
in
whereas 3 + 4 = 2 in
Cyclic groups and all their subgroups are abelian. Every element is of the form
then 
so every element commutes with every other.
If
then
for all
This is because
Ifis a cyclic group of orderthen every subgroup ofis cyclic. The order of any subgroup ofis a divisor ofand for each positive divisor
ofthe grouphas exactly one subgroup of order
Ifis finite, then there are exactly
elements that generate the group on their own, whereis the number of numbers in
that are coprime to
More generally, if
dividesthen the number of elements inwhich have orderis
The order of the residue class of m is
If
is prime, then the only group (up to isomorphism) with p elements is the cyclic group
or
The direct product of two cyclic groups
and
is cyclic if and only ifand
are coprime. Thus
is the direct product of
and
but not the direct product of
and
A primary cyclic group is a group of the form
whereis a prime number. The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many finite primary cyclic and infinite cyclic groups.
The elements of
coprime toform a group under multiplication modulowith elements,
When
we get
is cyclic if and only if
for
and
in which case every generator ofis called a primitive root moduloThus,is cyclic for
but not forwhere it is instead isomorphic to the Klein four-group.
The group
is cyclic with
elements for every prime
and is also written
because it consists of the non-zero elements. More generally, every finite subgroup of the multiplicative group of any field is cyclic.
