Let G be a group and g be an element of G. The smallest positive integer...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Let G be a group and g be an element of G. The smallest positive integer n such that g" = e is called the order of g. If such integer does not exist, then we say that g has infinite order. The order of g is denoted by g a) Assume that [g] =6. Show that g|-6 and |g³|= 2. b) More generally, if [g] =n, then for any nonzero integer k there is lg*1 = (mk) where (n, k) denote the greatest common divisor of n and k.. e) Show that the cyclic subgroup (g) of G, generated by the element g, has the order 19. Let G be a group and g be an element of G. The smallest positive integer n such that g" = e is called the order of g. If such integer does not exist, then we say that g has infinite order. The order of g is denoted by g a) Assume that [g] =6. Show that g|-6 and |g³|= 2. b) More generally, if [g] =n, then for any nonzero integer k there is lg*1 = (mk) where (n, k) denote the greatest common divisor of n and k.. e) Show that the cyclic subgroup (g) of G, generated by the element g, has the order 19.
Expert Answer:
Related Book For
Posted Date:
Students also viewed these mathematics questions
-
The group G contains a subgroup P of order 7 such that the smallest normal subgroup of G containing P is G itself. ex = x and (gig)x gi(gzx).
-
A college dean wants to determine the average time it takes a student to get from one class to the next. Preliminary studies have shown that it is reasonable to let o = 1.5 minutes (the standard...
-
Give at most a three-sentence synopsis of the proof of Theorem 6.6. 6.6 Theorem A subgroup of a cyclic group is cyclic. Proof Let G be a cyclic group generated by a and let II be a subgroup of G. If...
-
For nearly 50 years, Loganville Window Treatments (LWT) of Loganville, Georgia, has made interior shutters that are sold through decorating centers. Figure 3.20 shows some of the various styles of...
-
Give an example of a relation schema R and a set of dependencies such that R is in BCNF, but is not in 4NF.
-
Using the OVME screen for the options you evaluated in Exercise 1 , evaluate the prices of the options for different index prices. Click the "Scenario" tab and set the axis for option prices and...
-
Consider a publicly held company whose products you are familiar with. Some examples might include: Access the companys public website and search for its most recent annual report. (Some companies...
-
Harrison Forklift's pension expense includes a service cost of $10 million. Harrison began the year with a pension liability of $28 million (underfunded pension plan). Required: Prepare the...
-
A trader holds a long position (year 2020) of $1 MM of the 8 percent bond maturing 2040. The bond's modified duration is 11.14692, and it price is $129.87596.Its BPV is 0.14477.The trader decides to...
-
You are the owner of a small business that manages the program, food, and merchandise sales at the Excel Center, where the local professional basketball team plays. In addition to basketball games,...
-
The current I(t) in an LC series circuit is governed by the initial value problem below. Determine the current as a function of time t. 9 sint, Ost2x I''(t) + 41(t) = g(t). 1(0)=2. 1'(0) = 11, where...
-
Mr. Sho operates a news-stand and sells newspaper - orders copies of the newspaper from the publisher at a cost of $4 per copy - sells at a retailing price of $7 -The demand can be described with a...
-
Problem 1. Consider the Fibonacci numbers, define recursively by F = 0, F = 1, and Fn = Fn1 + Fn-2 for all n 2; so the first few terms are 0, 1, 1, 2, 3, 5, 8, 13, .... Fn For all n >2, define the...
-
On January 1, 2021, Entity A Company acquired all the assets and assumed all the liabilities of Entity B Company and merged Entity B into Entity A. In exchange for the net assets of Entity B, Entity...
-
You have been hired as a new fund manager by a large investment company that runs a number of mutual funds. You have been given $100 million to invest and have been assigned to start a new fund. It...
-
The deformation of a sheet of the metal unobtanium is recorded to be: 11 = 1.5 104 E12 2.5 x 10-5 E22 3.0 10-5 What are the volume and deviatoric strains of the sample? What are the principal strains...
-
Job cost sheets for incomplete jobs in Work in Process on February 1 showed the following: Job # 3 6 Job # 3 7 Direct materials $ 4 8 0 $ 3 9 0 Direct labor 5 6 0 3 4 0 Manufacturing overhead 8 4 0 5...
-
Pappa's Appliances uses the periodic inventory system. Details regarding the inventory of appliances at January 1, purchases invoices during the year, and the inventory count at December 31 are...
-
Show that : C M 2 (R) given by for a, b R gives an isomorphism of C with the subring [C] of M 2 (R). p(a + bi) = D -b a
-
How many different homomorphisms are there of a free group of rank 2 onto a. Z 4 ? b. Z 6 ? c. S 3 ?
-
Example 25.12 described an ordering of Z[2] = {m + n2 |m, n Z} in which -2 is positive. Describe, in terms of m and n, all positive elements of Z[2] in that ordering. Data from Example 25.12...
-
The following summary data are from a performance report for Sterling Company for May, during which 9,600 units were produced. The budget reflects the companys normal capacity of 10,000 units. a....
-
Graham Corporation used the following data to evaluate its current Kin operating system. The company sells items for \($10\) each and used a budgeted selling price of \($10\) per unit. a. Prepare the...
-
Selected data for Miller Company, which operates three departments, follow: During the year, the companys fixed expenses included the following: Assume that the property tax rate is the same for both...
Study smarter with the SolutionInn App