Let G be a group of order 231 = 3 · 7 · 11. Let H, K and N
denote sylow 3,7 and 11-subgroups of G, respectively.
a) Prove that K, N are both proper subsets of G.
b) Prove that G = HKN.
c) Prove that N ≤ Z(G). (you may find below problem useful).
a): <|/ is a normal subgroup, i.e. K,N are normal subgroups
of G
(below problem): Let G be a group, with H ≤ G...
#11
11. If a group G has exactly one subgroup H of order k, prove that H is normal in G. Define the centralizer of an element g in a group G to be the set 12.
11. Prove that a nonempty subset H of a group G is a subgroup of G if and only if whenever a, b E H, then ab-1 e H
Let r(x) = f(g(h(z))), where h(1) = 2, 9(2) = 2, H' (1) = 9,9 (2) = -2, and f'(2) = 5. Find (1). 5 6 Answer:. '(1) = 0 11 12
3. 11,7,3,-1,-5-4-9-13-11-2 a) arithmetic b) a = 11 + (8 - 1) = 4 c) -21 10.4 For questions 1-5, a. Determine if the sequence is arithmetic or geometric. b. Write a formula for the nth term in the sequence. c. Find the 9th term in the sequence, using the formula from part b.
Year Return 1 9% 2 11% 3 -4% 4 7% 5 6% A) What is the geometric average? What is the arithmetic average? B) Assume that the distribution is normal. Using the standard error, calculate the 95.4 percent confidence interval for the true return.
Question 6 The following 4 questions are based on the following common SAMPLE: 3 5 5 6 9 3 What is the difference between the Arithmetic mean and the Geometric mean? O b. 1.23 O c. The correct answer does not appear as one of the choices
19. Let H and K be subgroups of a group G, where H = 9, 1K1 =12 and where the index [G:HNK] =IGI. Find (HNKI.
5. Show that 9: R - Z defined by g(x) = to AND that h: Z - Z given by h (2 one-to-one. (The "brackets" represent the ceiling function.) 6. Use the formula for the sum of the first n terms of a geometric sequence to write the following as a closed formula: 2.524+1 ke
3. Prove that (f = 0()] ^ [9 = 0(h)) = f =0(h).