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Section 4.1 Cyclic groups

Often a subgroup will depend entirely on a single element of the group; that is, knowing that particular element will allow us to compute any other element in the subgroup.

Example 4.1.1. An Infinite Cyclic Subgroup, Modular Addition.

Suppose that we consider 3Z and look at all multiples (both positive and negative) of 3. As a set, this is
3Z={,3,0,3,6,}.
It is easy to see that 3Z is a subgroup of the integers. This subgroup is completely determined by the element 3 since we can obtain all of the other elements of the group by taking multiples of 3. Every element in the subgroup is “generated” by 3.

Example 4.1.2. An Infinite Cyclic Subgroup, Multiplication of Rational Numbers.

If H={2n:nZ}, then H is a subgroup of the multiplicative group of nonzero rational numbers, Q. If a=2m and b=2n are in H, then ab1=2m2n=2mn is also in H. By Proposition 3.3.8, H is a subgroup of Q determined by the element 2.

Proof.

The identity is in a since a0=e. If g and h are any two elements in a, then by the definition of a we can write g=am and h=an for some integers m and n. So gh=aman=am+n is again in a. Finally, if g=an in a, then the inverse g1=an is also in a. Clearly, any subgroup H of G containing a must contain all the powers of a by closure; hence, H contains a. Therefore, a is the smallest subgroup of G containing a.

Remark 4.1.4.

If we are using the “+” notation, as in the case of the integers under addition, we write a={na:nZ}.
For aG, we call a the cyclic subgroup generated by a. If G contains some element a such that G=a, then G is a cyclic group. In this case a is a generator of G. If a is an element of a group G, we define the order of a to be the smallest positive integer n such that an=e, and we write |a|=n. If there is no such integer n, we say that the order of a is infinite and write |a|= to denote the order of a.

Example 4.1.5. Generators of a Finite Cyclic Group.

Notice that a cyclic group can have more than a single generator. Both 1 and 5 generate Z6; hence, Z6 is a cyclic group. Not every element in a cyclic group is necessarily a generator of the group. The order of 2Z6 is 3. The cyclic subgroup generated by 2 is 2={0,2,4}.
The groups Z and Zn are cyclic groups. The elements 1 and 1 are generators for Z. We can certainly generate Zn with 1 although there may be other generators of Zn, as in the case of Z6.

Example 4.1.6. A Cyclic Group of Units.

The group of units, U(9), in Z9 is a cyclic group. As a set, U(9) is {1,2,4,5,7,8}. The element 2 is a generator for U(9) since
21=222=423=824=725=526=1.

Example 4.1.7. A Group That is Not Cyclic.

Not every group is a cyclic group. Consider the symmetry group of an equilateral triangle S3. The subgroups of S3 are shown in Figure 4.1.8. Notice that every subgroup is cyclic; however, no single element generates the entire group.
Figure 4.1.8. Subgroups of S3

Proof.

Let G be a cyclic group and aG be a generator for G. If g and h are in G, then they can be written as powers of a, say g=ar and h=as. Since
gh=aras=ar+s=as+r=asar=hg,
G is abelian.