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

454 ALGEBRAIC SYSTEMS [APP. B

B.9.Letσandτbe the following elements of the symmetric groupS 6 :

σ=

(

123456

315462

)

and τ=

(

123456

531624

)

Find:τσ,στ,σ^2 , andσ−^1. (Sinceσandτare functions,τσmeans applyσand thenτ.)

Figure B-7 shows the effect on 1, 2, ..., 6 of the composition of the permutations:

(a) σand thenτ;(b)τand thenσ;(c)σand thenσ, i.e.σ^2.

Thus:

τσ=

(

123456

152643

)

,στ=

(

123456

653214

)

,σ^2 =

(

123456

536421

)

We obtainσ−^1 by interchanging the top and bottom rows ofσand then rearranging:

σ−^1 =

(

315462

123456

)

=

(

123456

261435

)

Fig. B-7

B.10. LetHandKbe groups.

(a) Define the direct productG=H×KofHandK.

(b) What is the identity element and the order ofG=H×K?

(c) Describe and find the multiplication table of the groupG=Z 2 ×Z 2.

(a) LetG=H×K, the Cartesian product ofHandK, with the operation∗defined componentwise by

(h, k)∗h′,k′=(hh′,kk′)

ThenGis a group (Problem B.68), called thedirect productofHandK.

(b) The elemente=(eH,eK)is the identity element ofG, and|G|=|H|·|K|.

(c) SinceZ 2 has two elements,Ghas four elements. Let

e=( 0 , 0 ), a=( 1 , 0 ), b=( 0 , 1 ), c=( 1 , 1 )

The multiplication table ofGappears in Fig. B-8(a). Note thatGis abelian since the table is symmetric. Also,

a^2 =e,b^2 =e,c^2 =e. ThusGis not cyclic, and henceG∼=Z 4.

B.11. LetSbe the square in the planeR^2 pictured in Fig. B-8(b), with its center at the origin 0. Note that the
vertices ofSare numbered counterclockwise from 1 to 4.

(a) Define the groupGof symmetries ofS.

(b) List the elements ofG.

(c) Find a minimum set of generators ofG.