Let G' be the derived subgroup. Prove: a) If G' is a subgroup of H and H is a subgroup of G, show that H is normal in G; b) If K is a normal subgroup of G, then K' is a normal subgroup of G c) Suppose f:G --> H is an epimorphism, with ker(f)=K, then H is abelian iff G' is a subgroup of K. I need a detailed proof of t
Let G be a group, not necessarily finite, and let H be subgroup G. (a) Prove that U = intersection of all x in G xHx^-1 is the largest normal subgroup of G contained in H. (b) Show that no proper subgroup H of A_5 contains six distinct Sylow 5-subgroups. I need a detailed rigorous proof of this to study please.
If |G|=p^n, where p is a prime, show that G has subgroups G_0, G_1, . . ., G_n with 1= G_0 <= G_1 <= . . . <= G_n =G such that [G_i : G_{i-1}] = p, 1 <= i <= n. I need a rigorous, detailed proof of this to study please.
Let A and B be finite subgroups of G. Even though AB need not be a subgroup of G, show that |AB||A n B| = |A||B|. (Hint: Define (a_1,b_1)~(a_2,b_2) iff a_1b_1 = a_2b_2. Prove that ~ is an equivalence relation and examine the equivalence classes.) I need a detailed and rigorous proof to study for a test p
Show that {a sub n} infinity. Converges to A iff n=1 {a sub n - A}infinity. Converges to 0 n=1. .
A set F is closed iff for all x_n in F for all n in the naturals (x_n) converges to l, then l is in F. (x_n) is a sequence
Survey of 36 students Students' Preference: 25 Cake 20 Ice Cream 15 pie 2 all three 1 no desserts 15 cake or ice cream 8 pie or cake 3 ice cream only Draw a Venn Diagram that will represent the responses.
1. (3 points) - If A  B and B  C, can you conclude that A  C? Can you conclude that A  C? Definitions of symbols: ⊂ "is a proper subset of" ⊆ "is a subset of" ⊄ "is not a subset of" 1) Yes. All of the elements in A are in B, and all of the elements in B are in
Hi, I need help with this assignment. I will be using this to check my work, so it very important that all work is shown so that I have a clear idea of how each problem is done. I am only interested in the completed assignment- not just some of it. Thank you very much for your help! :) (Please see the attached file) Instruct
Prove: A group G is the union of 3 proper subgroups iff there exists an epimorphism between G and the Klein 4-group.
Use the fact that any finitely generated subgroup of (Q,+), (the rationals under addition), is cyclic to show that (Q,+)+(Q,+), (the direct sum of two copies of (Q,+)), is not isomorphic to (Q,+). The first part of this was showing that any finitely generated subgroup of (Q,+) is cyclic and I understand that part and really had
Two movie theaters, UAI and UAII, start their movies at 7:00 p.m. The movie at UAI takes 75 minutes and the movie at UAII takes 90 minutes. If the shows run continuously, when will they again start at the same time?
Retirement Annuity - When she is 30, Sue starts making annual deposits of $2,000 into a bondfund that pays 8% annual interest compounded continuously. Assuming that her deposits are made as a contiuous income flow, how much money will be in her account if she retires at age 55?
Solve this problem two different ways: Mensa is an association for people with high IQs. An advertisement for the organization offers the following challenge: "Take the instant test to see if you are a genius." PUT THE APPROPRIATE PLUS OR MINUS SIGNS BETWEEN THE NUMBERS, IN THE CORRECT PLACES, SO THAT THE SUM TOTAL WIL
1. Determine without graphing whether the given quadratic function has minimum value or maximum value. Then find the coordinates ... 2. Solve the Triangle. Round lengths ... 3. Solve the problem: One number is 6 less than a second number. Twice the second number is ... 4. Find the standard form of the equation of the el
In the textile industry, filament and yarn sizes are reported in denier which is defined as the mass in grams of a length of 9000 meters. If a synthetic fiber has an average specific gravity of 1.32 and a filament of this material has a denier of 5.0, what is the mass per unit length in pounds-mass per yard? What is the cross se
Calculate the ratio of the area to the volume for a unit cube, a unit sphere inscribed inside the cube, and a right cylinder inscribed inside the cube. Next, for each having a unit volume (i.e., all three solids have the same volume) calculate the area-to-volume ratios for a sphere, cube, and cylinder.
Suppose that we wished to develop a base three number system to be used in place of either a decimal system or a base two binary system a-- How many separate symbols would we need in this system? In the binary system, we can get away with only two, 0 and 1. Will this work in a base three system? How would you write the decim
Question 1.The world record for the men's long jump is 29 3/8 feet. The women's record for the same event is 24 33/48 feet. How many feet longer is the men's record? Question 2. Each lap of the Indianapolis 500, a 500-mile car race, constitutes 1/200 of the race. How many miles are in a lap? Question 3. In a survey of fav
What is the algorithm for computing the sum of an arithmetic progression of n terms with first term a and common difference d? -------------- Step 1 Set S = a, k = 1, and t = a. Step 2 while k < n (a) Replace t with t + d (b) Replace S with S + 1 (b) Replace k with k + 1
Let d be the usual Euclidean metric on R^n and f:R^n -> R^n be any function satisfying d(f(x),f(y))<d(x,y) for all distinct x,y in R^n. If B is a bounded subset of R^n such that f(B) is contained in B, show there is a unique b in the closure of B such that f(b) = b. I can show the uniqueness of such a b. Please give a detailed s
Need solution for the 2 problems attached. Find the solution of each of the following (see attached file).
PROVE THAT 5 * 7^n + 3 * 11^n is divisible by 4 for all integers n >=0.
A set S of bit strings is defined in the following way: 1. the empty string is an element of S 2. 0x1 is an element of S if x is an element of S. a. List five specific elements of S. b. List strings, if any, of length six which are elements of S. c. Prove or disprove that there are no strings of length eleven is S.
A. You are given a list of n names (all of the names are distinct) with instructions to put them in alphabetical order. How many comparisons are necessary to accomplish your task? B. After you have accomplished the task to part a, you are given another name. How many comparisons are necessary to place this name in its place i
Let M be a two by two matrix for which each element is a real number, and let S be the set of all such matrices. Consider the mapping f of S to the real numbers R defined by the relation f(M) = determinant of M. a. Is this mapping onto? Why or why not? b. Is this mapping one to one? Why or why not? c. Is this mapping a h
When three pumps, A, B, and C run together, they pump 3700 gallons/hour. When only A and B are running, 2700 gals/hr is the result and when only A and C are running, 2400 gals/hr is the result. What is the pumping capacity of the C pump, in gals/hr?
I am trying to work these problems not quite sure of how to work them. Find the equation of the line shown ... A bank loaned $15,000, some at an annual rate of 16% and some at an annual rate of 10%. If the income from these loans was $1800, how much was loaned at 10%? Write the augmented matrix of the system ... 2x
prove 1 a/b=(ac)/(bc),c can't equal 0 2 a-b < 0, "if and only if" a < b 3 if a < b then -b < -a 4 if 0 < a < b, then (1/b) < (1/a 5 a^2 greater than or equal to 0
-(-a)=a -(a)*-(b)=a*b if a doesn't equal 0 then 1/a doesn't equal 0 if a doesn't equal 0 then (-1)/a = 1/(-a) a(b-c)=ab-ac