Group Theory Worksheet

Transcription

1 Jonathan Loss Group Theory Worsheet Goals: To ntroduce the student to the bascs of group theory. To provde a hstorcal framewor n whch to learn. To understand the usefulness of Cayley tables. To specfcally examne the group of the quaternons. Target Audence: Ths worsheet s amed at provdng an ntroducton to the bascs of group theory. It s best suted for those students, whether n hgh school or undergraduate studes, who have been exposed to set notaton and modular arthmetc. There are bref revews of these operatons, and defntons of all maor terms are gven; yet pror exposure s helpful.

2 Group Theory Worsheet 2 Introducton: Group theory s a part of a larger branch of mathematcs nown as abstract or modern algebra. Ths branch s concerned wth dfferent algebrac structures and how they nteract wth each other more than t s concerned wth what the actual elements are. Some other examples of algebrac structures are rngs, felds, ntegral domans, and vector spaces. Group theory was the frst of these algebrac structures to be avdly studed. Its roots go bac to the 3 th and the 5 th centures when the Moors and Leonardo da Vnc studed symmetry ndependently of each other. Then, n the early 700s Joseph Lagrange studed permutatons of elements (how many dfferent ways the elements of a set could be arranged), but group theory was frmly establshed as ts own feld of study n the md 800s. Evarste Galos frst used the word group n ts techncal sense n a paper he publshed n 830. In order to go deeper nto understandng the subect, t would be best to gve a formal defnton of a group at ths pont: Defnton: A group G s a nonempty set of elements under a bnary operaton * for whch the followng four condtons hold: ) G s closed under the operaton * : ( a * b) ΠG when a, b ΠG. 2) The operaton * s assocatve: a *( b * c) = ( a * b) * c " a, b, c ΠG. 3) G has an dentty element e: a * e = e * a = a " a ΠG. 4) Each element of G has an nverse: a * b = b * a = e. A formal axomatc defnton, smlar to the one above, was frst gven n a paper publshed n 882 by Henrch Weber. Untl that pont, mathematcans had been actvely worng wth the concepts of groups, but no one had formally set down a defnton. Examples of Groups: It would be benefcal to gve a few examples of groups so that the reader may better comprehend the defnton gven above. Example : The set of ntegers ( Z ) under the operaton of addton s a group. Ths set s closed: pc any two ntegers a and b, and ther sum a + b s also an element of the ntegers. The set s assocatve snce addton on the ntegers s assocatve. The dentty element s 0. And for any element a, ts nverse s a. Therefore t satsfes all the requrements to be a group. Before proceedng to Example 2, a short revew of congruence classes wll be dscussed. Remember that x y mod n f and only f x y s a multple of n. For example, 2 6 mod 2 because 2 6 = 6 and 6 = 3 2. Also, ( 5 8) 5mod 7 because 5 8 = 40 and 40-5 = 35 = 5 7. You wll need ths nowledge to answer the questons n Example 2. 5 under multplcaton mod 40 s a group. The product of any two elements n the set s also a member of the set. Assocatvty holds snce multplcaton modulo n s assocatve. Example 2: The set of elements {,5,25,35}

3 Group Theory Worsheet 3 What s the dentty element n ths set? What are the nverses of each of the four elements? Example 3: The set of ntegers under subtracton s not a group. Gve an example to llustrate why t does not satsfy property 2 of a group. Gve an example to llustrate why t does not satsfy property 3. Example 4: The set of ntegers under multplcaton s not a group. Why not? Order of a Group: Another mportant defnton s that of the order of a group. Knowng the order of a group s very useful when worng wth more advanced group theory, yet t s a very smple concept to understand. Defnton: The order of a group G, denoted o(g) or G, s the number of elements that the group contans. If G does not have a fnte number of elements, ts order s sad to be nfnte. Example : The set = { 5,5,25,35} A under multplcaton mod 40 has order 4, denoted o(a) = 4. Example 2: The set of ntegers under addton has nfnte order because t does not have a fnte number of elements. Cayley Tables: One of the ntegral founders of group theory (and abstract algebra n general) was Arthur Cayley (82895). He performed much-needed research n the feld that led to the development of what s nown today as Cayley s Theorem. The detals of that theorem are beyond the scope of ths exercse, but he also contrbuted n another area of group theory that s useful to us here. Cayley organzed the elements of a group along the top and left hand sdes of a table. He then flled n the mddle wth the result of a certan operaton performed on two elements. For example, consder the set of complex numbers G = {,,, } under multplcaton. Cayley would have descrbed ther nteracton through a table smlar to the one that follows:

4 Group Theory Worsheet Ths Cayley table dsplays all the possble products when two elements are multpled together. It s read as an element on the left column tmes an element on the top row. Notce, n ths case t does not matter whch way you thn of dong the multplcaton. The top tmes the left wll equal the left tmes the top. Ths s because multplcaton n ths group s commutatve;.e., ab = ba " a, b Œ G. However, ths s not true n all groups. (Groups that are commutatve are also nown as Abelan groups, after the Norwegan mathematcan Nels Abel who wored wth the theory of equatons n the early 800s.) Fll n the followng Cayley table based on the group = { 5,5,25,35} modulo 40: G under multplcaton Is ths group Abelan? The Quaternons: A specal group that stands out for ts mportance to group theory s that of the quaternons, { ±, ± ±, ± }. Wllam Hamlton dscovered ths group n 843 after extensve research. Ths group s used to extend the complex numbers nto three and four dmensons, but ts most famous qualty s one that we wll dscover later n ths exercse. An nterestng account of how Hamlton dscovered the quaternons s found n a letter he wrote to hs son many years after the dscovery (the excerpt that follows was taen from Gallan, 2002, p. 57): But on the 6 th day of the same month [October 843] whch happened to be a Monday and a Councl day of the Royal Irsh Academy I was walng to attend and presde, and your mother was walng wth me, along the Royal Canal, to whch she had perhaps been drven; and although she taled wth me now and then, yet an under-current of thought was gong on n my mnd, whch gave at last a result, whereof t s not too much to say that I felt at once the mportance. An electrc crcut seemed to close; and a spar flashed forth, the herald (as I foresaw mmedately) of many long years to come of defntely drected thought and wor, by myself f spared, and at all events on the part of others, f I should ever be allowed to lve long enough dstnctly to communcate the dscovery. I pulled out on the spot a pocet-boo, whch stll exsts, and made an entry there and then. Nor could I resst

5 Group Theory Worsheet 5 the mpulse unphlosophcal as t may have been to cut wth a nfe on a stone of Brougham Brdge, as we passed t, the fundamental formula wth the symbols,, ; 2 = 2 = 2 = =, whch contans the soluton of the Problem, but of course as an nscrpton, has long snce mouldered away. Hamlton, n hs letter above, denoted one of the basc propertes of ths group: the fact that squarng any of the elements,, or, wll produce the number. Also, = = -, = = -, = = -. It s helpful to thn of the multplcaton of these elements by usng the followng crcle: Gong n a clocwse manner, multplyng two consecutve elements yelds the thrd element. Gong counter-clocwse yelds the negatve of the thrd element. The followng Cayley table has been partally flled n. Use your nowledge of the multplcaton n the quaternons n order to fll n the remanng entres What s the order of the quaternons? Is ths group Abelan? If not, gve an example. You have ust dscovered one of the most mportant propertes of ths group. It s non-abelan! Ths s noteworthy, for t was the frst rng (an extenson of the dea of a group) to be dscovered that was non-commutatve.

6 Group Theory Worsheet 6 References: Two textboos were extremely useful n the creatng of ths worsheet. They helped foster deas for examples and ncreased my nowledge on the subect tself. They are as follows: Gallan, J.A. (2002). Contemporary Abstract Algebra (5 th ed.). New Yor: Houghton Mffln. Glbert, J. & L. Glbert. (2000). Elements of Modern Algebra (5 th ed.). Pacfc Grove: Broos/Cole.