## 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 is itself suffices to show that is 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 If is a cyclic group of order then every subgroup of is cyclic. The order of any subgroup of is a divisor of and for each positive divisor of the group has exactly one subgroup of order If is finite, then there are exactly elements that generate the group on their own, where is the number of numbers in that are coprime to More generally, if divides then the number of elements in which have order is 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 if and 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 where is 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 to form a group under multiplication modulo with elements, When we get  is cyclic if and only if for and in which case every generator of is called a primitive root modulo Thus, is cyclic for but not for where 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. 