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 groupthat containsisitself suffices to show thatis cyclic.

For example, ifis a group, thenand G is cyclic. In fact,is isomorphic towith additionFor example, corresponds toWe can use the isomorphismdefined by

For every positive integerthere is exactly one cyclic group (up to isomorphism) whose order isand 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 denotedor or C-n where n is the order, equal to the number of elements.

inwhereas 3 + 4 = 2 in

Cyclic groups and all their subgroups are abelian. Every element is of the formthen so every element commutes with every other.

Ifthenfor allThis is because

Ifis a cyclic group of orderthen every subgroup ofis cyclic. The order of any subgroup ofis a divisor ofand for each positive divisorofthe grouphas exactly one subgroup of order

Ifis finite, then there are exactlyelements that generate the group on their own, whereis the number of numbers inthat are coprime toMore generally, ifdividesthen the number of elements inwhich have orderisThe order of the residue class of m is

Ifis prime, then the only group (up to isomorphism) with p elements is the cyclic groupor

The direct product of two cyclic groupsandis cyclic if and only ifand are coprime. Thusis the direct product ofandbut not the direct product ofand

A primary cyclic group is a group of the formwhereis 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  ofcoprime toform a group under multiplication modulowith elements,Whenwe get

is cyclic if and only ifforandin 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 withelements for every primeand 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.