New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

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

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

The Presidency

by: Boris Kuhn

The Presidency POLI 100A

Boris Kuhn

GPA 3.81


Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

Class Notes
25 ?




Popular in Course

Popular in Political Science

This 2 page Class Notes was uploaded by Boris Kuhn on Thursday October 22, 2015. The Class Notes belongs to POLI 100A at University of California - San Diego taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/226802/poli-100a-university-of-california-san-diego in Political Science at University of California - San Diego.

Popular in Political Science


Reviews for The Presidency


Report this Material


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: 10/22/15
Subgroups of Z and of Zquot Math 100 I39CSI likall 2001 In this handout wo 39lassily tho subgroups of Z and Z IVor oyory intogor n nZ is a subgroup of Z and an nk L E Z is a subgroup of Z l or oxamplo tho subgroup of Zm additiyoly gonoratod by l is 42l0 4812160 48260 02468 Theorem 1 liirrr y N llbfquotUllp of Z has UM form nZ for I unitm HIM 1 I Z 0 Proof 39l ho main idoa is this in tho subgroup nZ for II gt 0 n is tho loast positiyo olomont 39l horoloro if wo want to proyo an arbitrary nontriyial subgroup II C Z has tho form nZ wo should dolino n as tho loast positiyo olomont of II First wo nood to show II has a positiyo olomont at all Lot II C Z bo a subgroup ll II thon II nZ for n 0 ssumo now that II y so II 39ontains a nonxoro intogor say 1 Sim39o II is quotlosod undor nogation 1 E II Ono of 1 or 1 is positiyo so II 39ontains a positiyo intogor 39l horoloro II 39ontains a 4181 positiyo intogor say I V Vo will proyo II nZ Sim39o n E II quotlosuro of II undor addition and nogation implios nZ C II For tho royorso inlt39lusion pick I E II To show I E quotZ wo want to show Ill1 Lot39s uso tho division algorithm with division by I not by h whyE l or somo intogors and I39 I1 quot1 139 0 f I39 lt n Sim39o I E II and m 6 NZ C II wo hayo I39 I1 m E II 39l ho inoqualitios 0 f I39 lt n and tho minimality of n among tho positiyo intogors in II imply I39 0 39l horoloro I1 IN 6 NZ and wo hayo II nZ Convorning tho uniquonoss of II noto that within tho subgroup nZ for n gt 0 wo 39an quotharavtorim n as tho loast positiyo olomont 39l horoloro n is uniquoly dotorminod by nZ oyon whon II 0 for obyious roasons Corollary 1 51317 I pusftfirr HIM 1 m 39Trr subgroups ufZ N lls fylllfj mZ C II C Z In prrri39srly It N llifquotUllps IZ iiiMN Iml gt 0 Proof If 11m say In Mt thon mZ C IZ C Z Conyorsoly if mZ C II C Z thon wo know by 39l hoorom 1 that II nZ for somo II gt 0 Sim39o m E mZ C II nZ wo soo that m E quotZ so quotI ll VVrito n as I so II IZ and Irr1l gt 0 D Theorem 2 Fur filthy4 3 1 1an I all 1an only 12 3 bZ l Proof If 1I thon I II for an intogor It 39l hus IH39 II139 for any intogor I39 so bZ C IZ Conyorsoly if bZ C IZ thon I E Z is a multiplo of 1 say I II 39l hon 1 D In words tho thoorom says to liyido is to 39ontain39 Commit this phraso to momory so you don39t think 1 gt IZ C Z by mist o 39l ry out a lt onlt39roto oxamplo liko 2Z and SZ VVhilo 2 G tho multiplos of G aro multiplos of 2 not tho othor way around so GZ C 2Z y ho subgroups of Z using tho lt lassililt39ation of tho sulr Now wo 39 groups of Z Theorem 3 7 ma r m furr subgroup 0me has IIr form IZquot for I Mfrm Im I gt 0 In pm l llIIH In Zquot 1 Mid IIH r Is only our subgroup of HIPI pumi bh In ururI Is 1 llbfI U llp Is yrIr If yrIr l muf It is 39loar that for Ilm and I gt 0 IZquot is a subgroup of Z VVhon Ilm In E II mod 1 ltgt 1 E I mod InI so IZm IIII and thoroloro tho subgroups IZquot aro listinguishod by thoir sixo sim39o wo 39an road oll39 I from tho sixo IIII as m is of 39ourso k n ow n lVor oxamplo 2Z6 024 has sixo 32 3 Now wo show any subgroup of Zquot has tho losirod form Lot II C Zquot bo a subgroup VVo 39onsidor tho roduvtion homomorphism Z Z whilt39h sonltls 1 to H 1 mod 1 39l ho inyorso imago f IIII is a subgroup of Z whilt39h 39ontains f IIO IZ By Corollary 1 any subgroup of Z 39ontaining IZ has tho form IZ whoro Ilm and I gt 0 39l hat is IZ AII II E Z 1 mod 1 E II Sim39o is Slll j 391i If III II and thus II IZ IZm El Noto ospolt39ially that thoro is a onoitoiono 39orrospondonlt39o botwoon tho subgroups of Zquot and tho subgroups of Z intormodiato botwoon IZ and ting with an Z In a sonso tho way wo got tho subgroups of Zquot is by inlt39lusion liko IZ C IZ C Z and rodm39ing39 all groups modulo 1 to got 0 C IZquot C Z So tho subgroup stru39turo of Zquot is ossontially liko tho subgroup strult39turo of Z lying aboyo39 IZ lVor oxamplo tho subgroups of Z 39ontaining GZ aro SZ 32 22 Z


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

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

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Anthony Lee UC Santa Barbara

"I bought an awesome study guide, which helped me get an A in my Math 34B class this quarter!"

Steve Martinelli UC Los Angeles

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


"Their 'Elite Notetakers' are making over $1,200/month in sales by creating high quality content that helps their classmates in a time of need."

Become an Elite Notetaker and start selling your notes online!

Refund 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


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:

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

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.