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

APP. B] ALGEBRAIC SYSTEMS 441

EXAMPLE B.11

(a) Consider the permutation groupS 3 of degree 3 which is investigated above. The setH ={ε, σ 1 }is a

subgroup ofS 3. Its right and left cosets follow:

Right Cosets Left Cosets

H={ε, σ 1 } H={ε, σ 1 }

Hφ 1 ={φ 1 ,σ 2 } φ 1 H={φ 1 ,σ 3 }

Hφ 2 ={φ 2 ,σ 3 } φ 2 H={φ 2 ,σ 2 }

Observe that the right cosets and the left cosets are distinct; henceHis not a normal subgroup ofS 3.

(b) Consider the groupGof 2×2 matrices with rational entries and nonzero determinants. (See Example A.10.)

LetHbe the subset ofGconsisting of matrices whose upper-right entry is zero; that is, matrices of the form

[

a 0

cd

]

ThenHis a subgroup ofGsinceHis closed under multiplication and inverses andI∈H. However,His

not a normal subgroup since, for example, the following product does not belong toH:

[

12

13

]− 1 [

10

11

][

12

13

]

=

[

− 1 − 4

13

]

On the other hand, letKbe the subset ofGconsisting of matrices with determinant 1. One can show

thatKis also a subgroup ofG. Moreover, for any matrixXinGand any matrixAinK, we have

det(X−^1 AX)= 1

HenceX−^1 AXbelongs toK,soKis a normal subgroup ofG.

Integers Modulom

Consider the groupZof integers under addition. LetHdenote the multiples of 5, that is,

H={...,− 10 ,− 5 , 0 , 5 , 10 ,...}

ThenHis a subgroup (necessarily normal) ofZ. The cosets ofHinZappear in Fig. B-5(a). By the above
Theorem B.8,Z/H={ 0 , 1 , 2 , 3 , 4 }is a group under coset addition; its addition table appears in Fig. B-5(b).
ThisquotientgroupZ/Hisreferredtoastheintegersmodulo5anditisfrequentlydenotedbyZ 5. Analogously,
for any positive integern, there exists the quotient groupZncalled theintegers modulo n.

Fig. B-5