Question

Can you please help on Problem 3.10, Problem 3.13 and Problem 3.14 part 1 and 2 clearly. This is Abstract Algebra class and the topic is about Group d Hemak 3. A EGG d y or wewe both the the the widerstood. F will wie G.) www to - the bew a ble the wadog The wed ded The tity Exer 3.8. Expech of the group ppt is the w o man witve so De to i d eal cortly F ivity were ce of the request wie Vee Ghe G ha Remark . Note that p arties) ( and are both st a tements Sentine we want to prove these two propertis bold, we should be existence proofs ( els for all in there) Problemi G R -1) tabb e a standard wydation 1. Show that G ) is a binary structure 2. Show that G.-) group There are many other properties of that follow free at the base within the definition, we will find that we can prove many way s from theme of which we will see in the emple below and problems that follow. The proces are often very hot and simple, but i s require a bit of Example 11. Prove that a group hantly one Wette t is that there U for Proof El Ewry group has a wentity element, by definities So the westi part of this is only ) i ted that the shut the existence of Westy the way be structure in pista special of binary structure, the pool will applies It's very short, sowe Suppose les and are both times Now o r the producto. Since Meye But since bis an Identity , we also have ab Hence a Mary t w e have S e of the principles of e ntry alerta bere habitual that there without think I' to pay attention to some of the principles o f the bold in the contexto Remark 3.12. Working with mats with a group, the s o t the other tituted for the expense, where is any of the group which y o We all the riche platily because that's what it looks like. Similarly we place within the expe c t ed Notice that the procedures entre the special properties oa go they are dely as the fact that any part is a function erectly on with to Other t e The was the platio procedures de sobe with the grow. This way we tocca The t wo you to the I G group with a GP that the Path B 2. Suppose T serowot a tios table for group G. Use the best frien (1) to how the e is the below the ch o ppers at letre out the

Answer 1

Let then Cancelle e be the identity in (6 ) axe = a atetae - a 2 elita) = 0 [by the usuce either eto or ha=0 e-o became af - ☆ is commutative become att - atstab = btatta = br a = Lха - identity element en ls the e=0 L axe -a -eta e=0 is the & the inverse ich a - a -e = 6* a att tat = 0 7 6(Ha)=-a - Fa ['a##] i for at Ge we have a = - ta show (Gx) is a group. which associativity group J 3.13 G is a group. a, b, c to a it is given that ac =60- cer so é existo. right multipening in both side of 6 we have (ac) c = 6c cl alce = b(co-) L a group & ae = be re is the identity of the a) a = 6. os ae=a and be=6 . . Now let, ca=cb, now mw tipesing en in the left of with side have c(ca) = creo) (etc) a = (cc) [by associativity] g ea - eb o art