(a) Show that the additive group of rationals Q is not finitely generated. (b) Show that Q...

Question:

(a) Show that the additive group of rationals Q is not finitely generated.
(b) Show that Q is not free.
(c) Conclude that Exercise 9 is false if the hypothesis "finitely generated" is omitted.

Data from Exercise 9

Let G be a finitely generated abelian group in which no element (except 0) has finite order. Then G is a free abelian group.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Algebra Graduate Texts In Mathematics 73

ISBN: 9780387905181

8th Edition

Authors: Thomas W. Hungerford

See More Books

Question Posted: Apr 15, 2023 03:51 AM

See More Questions

KYC's stock price can go up by 15 percent every year, or down by 10 percent. Both outcomes are equally likely. The risk free rate is 5 percent, and the current stock price of KYC is 100. (a) Price a...
(a) Let G be the additive group of all polynomials in x with integer coefficients. Show that G is isomorphic to the group Q* of all positive rationaJs (under multiplication). (b) The group Q* is free...
Referring to Example 24.3, show that YX - XY = 1. Data from in 24.3 Example Let F be a field of characteristic zero, and let (F[x], +) be the additive group of the ring F[x] of polynomials with...
Refer to the following lease amortization schedule. The 10 payments are made annually starting with the inception Of the lease. Title does not transfer to the lessee and there is no bargain purchase...
Suppose the price of leather gloves is $12. Graph a demand curve for leather-glove laborers from the following data concerning their total physical product: QUANTITY OF LABOR TOTAL PHYSICAL PRODUCT...
es Ida Company produces a handcrafted musical instrument called a gamelan that is similar to a xylophone. The gamelans are sold for $737. Selected data for the company's operations last year follow:...
A six cylinder, four-stroke, spark ignition engine of bore $10 \mathrm{~cm}$ and stroke $12 \mathrm{~cm}$ with a compression ratio of 6 is tested at $4800 \mathrm{rpm}$ on a dynamometer of arm...
For the third quarter of the year, Johnson Company, 415 Circle Avenue, Chicago, Illinois 60652, received Form 941 from the Internal Revenue Service. The identification number of Johnson Company is...
A coffee filter of mass 1.2 grams dropped from a height of 3 m reaches the ground with a speed of 0.7 m/s. What is the change in thermal energy of the air and coffee filter? Start from the Energy...
Part One: In 2015, Fran Lexa opened Fran's Flowers, a small shop selling floral arrangements. On December 31, 2016, her accounting records show the following: Sales revenue...
Let G be a group containing an element a not of order 1 or 2. Show that G has a nonidentity automorphism.
A group G is nilpotent if and only if there is a normal series G = G o > G 1 > > G n = (e) such that G i /G i+1 < C(G/Gi+1) for every i.
A futures contract is used for hedging. Explain why the daily settlement of the contract can give rise to cash flow problems.
You're a manager in a large commercial bank. You discover that Patricia, a loan officer who reports to you, has forged an approval signature on a customer loan, which requires signatures from two...
The Seattle Real Estate Market : Data area vailable on a number of recent home sales in 3 different cities inthe Seattle, WA area. A regression model has been developed topredict selling price (in...
Your company is considering two mutually exclusive, equally 'risky, and not repeatable projects, A and B. Their cash flows are shown below. The CFO asked you to evaluate these two projects and is now...
Saudi Arabia follows Sharia Islamic law and has one of the lowest crime rates in the world. Punishment for crimes is severe--murderers are publicly beheaded and thieves have their hands amputated....
In the future year 20XX, the number of women senators and congressional representatives reached a critical mass to influence legislation affecting women. Through a coalition of Republican and...
Mr. LeClair and Ms. Bartoli own adjacent hobby farms. They have just received their property tax notices providing the following assessment and tax information: The regional government applies one...
A Firm intends to invest some capital for a period of 15 years; the Firm's Management considers three Options, each consisting of purchasing a machinery of a specific brand, different for each...
Implement the test in Problem 8.106, and provide a two-tailed p-value? Cardiovascular Disease Left ventricular mass (LVM) is an important risk factor for subsequent cardiovascular disease. A study is...
What test can be used to assess if there is a change in mean LVM over four years? Cardiovascular Disease Left ventricular mass (LVM) is an important risk factor for subsequent cardiovascular disease....
In a substudy, participants will be subdivided into two equal groups according to the severity of previous allergy symptoms (n = 30 each), and the effectiveness of the eye drop (vs. placebo) will be...
White Company has two departments, Cutting and Finishing. The company uses job-order costing and computes a predetermined overhead rate in each department. The Cutting Department bases its rate on...
3. Which line(s) in the main() will generate error? Explain your answer. #include using namespace std; class classP { public: int 11-1; protected: }; int 12 - 2; class classQ private classP{ public:...
Below are the account balances for Cowboy Law Firm at the end of December. Accounts Cash Salaries expense. Accounts payable Retained earnings Utilities expense Supplies Service revenue Common stock...

(a) Show that the additive group of rationals Q is not finitely generated. (b) Show that Q...

Question:

Step by Step Answer:

a To show that Q is not finitely generated we assume that Q is finitely generated by some elements a...View the full answer

Algebra Graduate Texts In Mathematics 73

Students also viewed these Mathematics questions