### Create a StudySoup account

#### Be part of our community, it's free to join!

Already have a StudySoup account? Login here

# Abstract Algebra MATH 3330

UH

GPA 3.69

### View Full Document

## 35

## 0

## Popular in Course

## Popular in Mathmatics

This 9 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 3330 at University of Houston taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/208414/math-3330-university-of-houston in Mathmatics at University of Houston.

## Reviews for Abstract Algebra

### What is Karma?

#### Karma is the currency of StudySoup.

#### You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 09/19/15

Section 31 Notes Page 1 of9 Math 3330 Section 31 Groups An algebraic system consists of a nonempty set With at least one equivalence relation and one or more binary operations We Will use the notation Set operation symbol operation symbol to denote an algebraic system So the integers With addition would be denoted Z T his is not used in the book but is much faster to type and more compact than the Written out version De nition A GROUP is an algebraic system consisting of am an Which Will be called multiplication but need NOT actually be multiplication With the following properties akin 6 Q 534cc we 104ml ABEL 1AA GKOUO Chuck 1 assocloiHVe 31W 0 1V1va 4 is ex Lbsuq PDTLssocioleE w mmm 0 fCi 13 77 O C osvyx asSocfoJ ive I O Ewe 96 opx 39s X o m Mu k OJ gran5 Section 31 Notes Page 2 of9 Example 4 the set of permutations on a nonempty set A With the group operation being function composition 4 A a A Sada Mu g is Iggec w LoSzIQ ore loi39eci t lt C amp3 J15 chciiw A550 clak wt hch 5 4 90Ai Giga ixm 1 a A 14 74 mam inverse i5 hijackve 321 exfsi s is 50 Lazc iim The permutation group on a set With three elements 3 1 M w uu bi ij n Y UJVJT Ilv vum S M es g y x 91719 f Section 31 Notes Page 3 of9 Cayley Table 7 a table With the group elements listed on the top row and rst column and the center of the table is the products of the elements Cayley Table for the symmetry group on 3 elements 0 Another Group 7 Let G i 1 i 1 With normal multiplication of complex numbers CayleyTable 0 Ccst oa I y 1 39 aQQ J W A L 115 0 Q l V A ul 39 Me a we 1 I A 39 a C2 Assoc X a350CIafl l39 I 4 4 u l A I m OMFY Mmlzy Is this group commutative Yb 5 Im II How do we tell from the table LB I71 Ve S LS v I quotUL L l I l a i The table below de nes a binary operation on the set Gamp ls n a group 15 km quot raw l d39blM 3 Does Q I heath if have m le SL quotlo cow C is We in emfy new rm M co W 4w COW NO am mi W m Wm Nc A G 40 UP Section 31 Notes Page 4 of9 De nitions If a group G has a nite number of elements it is called a nite group The number of elements of a nite group is called the order of the group If G does not have a nite number of elements it is called an in nite group FiniteGroupExamplei Q f 7 2 X In nite Group7 01 3V MuW or 91 qf 27 Theorem Properties of Group Elements Let G be a group With respect to a binary operation that is Written multiplicatively Then The identity element of G is unique 71 For every X E G 1ts 1nverse element X E G 1s unique 2 3 ForeveryXEG X7171 X X I2 C 4 Reverse order law For each X y E G xy 1 yilxil 5 Cancellation law If a X and y are in G and either of the equations aXayoryaXaaretruethenXy H 7 ex 39 Gf olx CC CDA x V07 Note the reverse order law 7 this is critical in nonabelian groups 7 6L I A y l 1 799 v39 x XV til 7 Y4 01 39gt 39 4 r 9 N c KY 11 Xy x LxvaXX c Maceme N ote 7 the cancellation law implies that each element only appears once in each row or column of the Cayley table of a group Mr a xy Pow leaIKE 0L M WA M Lamong cavvxb SW ColWW mm O eQwaML W W 0ms Qmm gtltm ya 3Xy 2ande 5 2x2 PageSof9 Mampr zcc A X 8 Equivalent Conditions for a Group 7 v 6 Let G be a nonempty set that is closed under an associative binary operation K 39 called multiplication Then G is a group if and only if the equations ax b and ya b have solutions X and y in G for all choices of a and b in G Section 31 Notes Proof 0 Assum G is a 9 0Lf Let Q 5 e c a 0539 e 6 39in axc ya 5 Claim k ave L y 7 bat I Know x 66 A E 6 CKX OLCa39IE ampQb 85 L Cym ba lltx Marray Le 5 quot6 Rsswm QOP Cu or I 6 here CKI39Sis X39elt o 532A W 6 xxzb ow Sivan G is o g C is CiOS39eoQ M A lS OxSSOC1Q iquotV meal Tberr 5 0M olenifty in G EVEy wech has FHA 5396 bk Rub Fix a I39m 6 K55 X 54 QXza 3A1 jstjc a 0v 7 5 0 Pig in Iar Se 9mg Lbi39 066 3 by gt ap39a s 7km X Zalx z ga zat o is eC rmarse 4 Section 31 Notes Page 6 of9 W512 Sam rick 0 Show 61 its EWL r39 fensc 0 Dig W G a X FigUr 39nVQ SC 17 let aneSC Xg Wt all Xy6 Extery 6W6 hots MVe S Pick a Q amp u a 0 t 6 F gigs afszlyota b 31 PE a Generalized Associativity The associative property is de ned for 3 elements abc abc If we ve a product with more than three elements how do we know we can regroup The answer is the Generalized Associativity Theorem It is proved in the book by complete induction where n represents the number of elements in the product The de nitions ofa group in this section and the properties of normal real number addition and multiplication give us a large number of groups to work with Including the following systems with Matrices menR menQ menZ menZn Invertible matrices in MIl with matrix multiplication a ZN 1 an COMKLL ind km grow0 Section 31 Notes Page 7 of9 What YOU need to be able to do with this material 7 I KNOW the de nition of a group and a commutative group 2 Take a system de ned for you and go through the steps to either prove it is a group or show why it is not a group Know the properties of a group element and know how to prove them L Example State the de nition of an additive group and state the properties of a group element in additive notation instead of multiplicative notation GI l3 is a rm0 IYC CD Cosqamp Valk Q Lio 399 6 Rssocimtiue Vqlbc 6 cu b Pa c at fbCgt IW Were exfsfs m elmWV E 5 G Suex VX 6 G X e c e F gtlt X out eO x020xgtlt LB I Ve Sej Vg amp 6 S39otoiA l AJ ant4030 OL C to Section 31 Notes Page 8 of9 Example 715 Z A a group When the operation A is de ned by x A y x y 1 forallxandyinZ ngedz alfo 65 QALJC 0E 6 2 A39s cfofx rKe Ab4c QLon Ac Cd39bwil c aloc42 Ab c7 amfbcz Oxf EHclD F O I V abc1 3 LW Leker FMamp 8 SG LM e XT X gtltquotegt 64X eXgt T l v I 019 Cor c E quot quot JA X X A X I 7 I DVW Sa 368 FinoQ w 6 Z Sow w X wa wa wgtlt 30km 43 LI A gt39Z x m kx gtlt14 gtltl w X AQ gt 1 X quotX Q Invvse o x 1395 39X 2 a 239 39rs ou Section 31 Notes Page 9 of9 Example 71s the set G 0 2 4 6 g Z10 a group with respect to multiplication multiplication mod 10 x 0 23 EU a If a O o 0 O O 3 O 1 2 a Q E i o 3 a Lf39 a CG 0 L Q g Z83 0 o a 5 quotL to 27 2 to EC M 3 3 7 Jews is 00746va EKQ Suing 0be gt 5 g Q 2quot 5 0t 3 S 395 M i5 caUCMQ SUE O V 066 t SKj sag0 m NW 020 is a 7 Q I15 6k 59570

### BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.

### You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

## Why people love StudySoup

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "Selling my MCAT study guides and notes has been a great source of side revenue while I'm in school. Some months I'm making over $500! Plus, it makes me happy knowing that I'm helping future med students with their MCAT."

#### "There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

#### "It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

### Refund Policy

#### STUDYSOUP CANCELLATION POLICY

All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email support@studysoup.com

#### STUDYSOUP REFUND POLICY

StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here: support@studysoup.com

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.