This is 2(b):
The following exercise shows that the converse to Lagrange's theorem is false, i.e. even if d ||G|, there need not be a subgroup of G with order d. (a) Let n > 4 and consider the alternating group An. Suppose that NC An is a normal subgroup and that there is a 3-cycle (abc) E N. Prove that N = An. Hint: it is enough to show that N contains all 3-cycles. What is the conjugate of...
Let a and b be elements of a group with identity 1. Suppose a and b relatively prime. Use Lagrange's Thm. to prove that (a)n (b)-(1} are
Prove or disprove the following. (a) R is a field. (b) There is
an additive identity for vectors in R^n. (If true, what is
it?)........
1. Prove or disprove the following. (a) R is a field (b) There is an it?) additive identity for vectors in R". (If true, what is (c) There is a is it? multiplicative identity for vectors in R". (If true, what (d) For , , (e) For a, bE R and E R", a(b) =...
If y = f(x), the inverse of f is given by Lagrange's identity: 1 dn-1 f-1(y) = = y + n! dyn–ī [y – f(y)]" when this series converges. (i) Verify Lagrange's identity when f (x) (ii) Show that one root of the equation x - 3x3 = i is = ac. 32n+1 (3n)! 2 = - (+) n!(2n + 1)! 43n+1 0 (iii) Find a solution for x, as a series in 1, of the equation x = eta
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and X1,X2, ,yn, and zi,22, An be real numbers. ,An, yī,Y2, #a) Prove the identity #b) Use the identity...
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in #a) to prove (the Cauchy-Schwartz inequality) that #1) Extend the result in #b) to prove that 4 #d) Use the inequality in #b) to prove the inequality which is the triangle inequality
#2. Let n E N and x1,x2,.., Xn, yı,y2,..,Ja, and zł,Zy, #a) Prove the identity An be real numbers #b) Use the identity in...
Prove the following vector identity using index notation A X (BXC) = (A.C)B - (A.B)C
Prove that conditional independence is symmetric (i.e. if A is independent of B given C then B is independent of A given C). Please type the answer for my notes thanks!
2. Use Lagrange's theorem to prove the Euler-Fermat Theorem: If n E Z+ and (a, n) = 1, then ap(n)-1 mod n.
Prove this identity