Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

442 ALGEBRAIC SYSTEMS [APP. B

Cyclic Subgroups

LetGbe any group and letabe any element ofG. As usual, we definea^0 =eandan+^1 =an·a. Clearly,
aman=am+nand(am)n=amn, for any integersmandn. LetSdenote the set of all the powers ofa; that is

S={···,a−^3 ,a−^2 ,a−^1 ,e,a,a^2 ,a^3 ,···}

ThenSis a subgroup ofGcalled the cyclic group generated bya. We denote this group bygp(a).
Furthermore, suppose that the powers ofaare not distinct, sayar=aswith, say,r>s. Thenar−s=e
wherer,s>0. The smallest positive integermsuch thatam=eis called theorderofaand it will be denoted
by|a|.If|a|=m, then the cyclic subgroupgp(a)hasmelements as follows:

gp(a)={e, a, a^2 ,a^3 ,...,am−^1 }

Consider, for example, the elementφ 1 in the symmetric groupS 3 discussed above. Then:

φ 11 =φ 1 ,φ 12 =φ 2 ,φ 13 =φ 2 ·φ 1 =e

Hence

∣

∣φ 1

∣

∣=3 andgp(φ 1 )={e, φ 1 ,φ 2 }. Observe that

∣

∣φ 1

∣

∣divides the order ofS 3. This is true in general;

that is, for any elementain a groupG,|a|equals the order ofgp(a)and hence|a|divides|G|by Lagrange’s
Theorem B.7. We also remark thatagroupGis said to becyclicif it has an elementasuch thatG=gp(a).

Generating Sets, Generators

Consider any subsetAof a groupG. Letgp(A)denote the set of all elementsxinGsuch thatxis equal to a
product of elements where each element comes from the setA∪A−^1 (whereA−^1 denotes the set of inverses of
elements ofA). That is,

gp(A)={x∈G

∣

∣x=b 1 b 2 ...bm where eachbi∈A∪A−^1 }

Thengp(A)is a subgroup ofGwithgenerating setA. In particular,Ais said to generate the groupGifG=gp(A),
that is, if everyginGis a product of elements fromA∪A−^1. We sayAis aminimal set of generatorsofGif
AgeneratesGand if no set with fewer elements thanAgeneratesG. For example, the permutationsa=σ 1 and
b=φ 1 form a minimal set of generators of the symmetric groupS 3 (Fig. B-4). Specifically,

e=a^2 ,σ 1 =a, σ 2 =ab, σ 3 =ab^2 ,φ 1 =b, φ 2 =b^2

andS 3 is not cyclic so it cannot be generated by one element.

Homomorphisms

A mappingffrom a groupGinto a groupG′is called a homomorphism if, for everya,b∈G,

f(ab)=f(a)f(b)

In addition, iffis one-to-one and onto, thenfis called anisomorphism; andGandG′are said to beisomorphic,
writtenG∼=G′.
Iff:G→G′is a homomorphism, then the kernel off, written Kerf, is the set of elements whose image
is the identity elemente′ofG′; that is,

Kerf={a∈G|f(a)=e′}

Recall that the image off, writtenf (G)or Imf, consists of the images of the elements underf; that is,

Imf={b∈G′|there existsa∈Gfor whichf(a)=b}.

The following theorem (proved in Problem B.19) is fundamental to group theory.