Throughout this question, let G be a finite group, let p be a prime, and suppose that H ≤ G is such that [G : H] = p.
Let G act on the set of left cosets of H in G by left multiplication (i.e., g · aH = (ga)H). Let K be the set of elements of G that fix every coset under this action; that is,

K = {g ∈ G : (∀a ∈ G) g · aH = aH}.

(a) Prove that K is normal subgroup of G and K⊆H. From the result of part (a) it follows that K ≤ H. For the remainder of this problem, we let k = [H : K].

(b) Prove that G/K is isomorphic to a subgroup of S_p.

(c) Prove that pk | p!. Hint: Calculate [G : K].

(d) Now suppose, in addition to the setup above, that p is the smallest prime dividing |G|. Prove that H is normal subgroup of G. Hint: Show that k = 1.

Friendly Co. produces two products, A and B. The sales volume for A is at least 80% of the total
sales of both A and B. However, the company cannot sell more than 110 units of A per day. Both
products use one raw material, of which the maximum daily availability is 300 kg. The usage
rates of the raw material are 2 kg per unit of A, and 4 kg per unit of B. The profit units for A and
B are $40 and $90, respectively.
Formulate the above problem as a linear programming problem. Then solve the problem to
determine the most optimal product mix for the company.

Q1.a) Assume known that 857 and 503 are primes. Determine whether 503 is a quadratic residue modulo 857 or not.

b) Slove the second-degree congruence equation 2x²+ 3x-11congruent (mod 31)

c) calculate the remainder when 7³³³³+ 251617⁸ is divided by 11

Assume that wt ∼ wn(0, σ2 ), t = 1, 2, . . . ,. Define Yt = µ + wt − θwt−1, where µ and θ are constants. Find E(Yt), var(Yt), and cov(Yt , Yt+k), where k is an integer and k ≥ 0

A group of k Vikings independently set out to make a new home. Each Viking has a copy of the same map, showing n islands. Each Viking decides to set sail for some random island. If two or more Vikings land on the same island, they have a battle. (No matter how many Vikings land on that island, it counts as one battle.)

(a) How many battles do we expect will occur? (Hint: Fix a single island, what is the probability of no viking ever landing there? What about exactly one viking reaching this island? What is the relationship of these events and there being a fight in the island?)

(b) You should have obtained a closed formula that depends on n and k. For both formulas, consider the cases in which there is only one island on the map. Do your solutions confirm the intuitive answer for this case? What if there’s only one Viking? What answers do you get for 400 Vikings and 100 islands?

Consider the following recurrence relation defined only for n = 2^k for integers k such that k ≥ 1: T(2) = 7, and for n ≥ 4, T(n) = n + T(n / 2). Three students were working together in a study group and came up with this answer for this recurrence: T(n) = n * log2 (n) − n − log2 (n) + 8. Determine if this solution is correct by trying to prove it is correct by induction.

How to prove that player I can always win a Nim game in which hte number of heaps with an odd number of coins is odd.

I understand the objective, having trouble formuating a more formula argument for the problem.

Show that if there are 100 people of different heights standing in a line, then it is possible to find at least 10 people in the order they stand in the line with increasing heights, or at least 12 people with decreasing heights.

Can you write an essay on the question below?

Give a critical analysis of the statement, “Mathematical inquiry is a gift to humankind from the mind of God.”

Prove that Sn is generated by {(i i + 1) I 1 < i < n - 1}.

1-Use the​ (x,y) coordinates in the figure to find the value of the trigonometric function at the indicated real​ number, t, or state that the expression is undefined.

SIN 11π/6

2-Use the​ (x,y) coordinates in the figure to find the value of the trigonometric function at the indicated real​ number, t, or state that the expression is undefined.

COS π//6

3- Use the​ (x,y) coordinates in the figure to find the value of tan 4π/3

or state that the expression is undefined.

4-Use the​ (x,y) coordinates in the given figure to find the value of the trigonometric function at the indicated real​ number, t, or state that the expression is undefined.

CSC 7π/6

5-Use the unit circle to find the value of COS 2π

and even or odd trigonometric functions to find the value of COS(-2π)

6- Use the unit circle to find the value of sin 7π/6

and even or odd trigonometric functions to find the value of sin (-7π/6)

7- Use the unit circle to find the value of tan 2π/3

and even or odd trigonometric functions to find the value of tan (-2π/3)

8-Use the unit circle to find the value of cos3π/2

and periodic properties of trigonometric functions to find the value of cos 11π/2

9- Let sin t =a​, cos t =b​, and tan t =c. Write the expression in terms of​ a, b, and c

6 sin( -t ) - sint t

10-  Let sin t =a​, cos t =b​, and tan t =c. Write the expression in terms of​ a, b, and c

sin(t+6π)−cos(t+4π)+tan(t+7π)=

This is Using MATLAB:

I am trying to store the solution of this matrix because I want to do something with the result like find the norm of that answer however I am stuck and cannot seem to be able to. Help would be appreciated!

---------------------------------------------

MATLAB CODE:

close all

clear

clc

A = [1 -1 2 -1;

2 -2 2 -3;

1 1 1 0;

1 -1 4 5];

b = [-8 -20 -2 4]';

x = gauss_elim(A,b)

function x = gauss_elim(A, b)

[nrow, ~] = size(A);

nb = length(b);

x = zeros(1,nrow);

% Gaussian elimination

for i = 1:nrow-1

if A(i,i) == 0

t = min(find(A(i+1:nrow,i) ~= 0) + i);

if isempty(t)

disp ('Error: A matrix is singular');

return

end

temp = A(i,:); tb = b(i);

A(i,:) = A(t,:); b(i) = b(t);

A(t,:) = temp; b(t) = tb;

end

for j = i+1:nrow

m = -A(j,i) / A(i,i);

A(j,i) = 0;

A(j,i+1:nrow) = A(j,i+1:nrow) + m*A(i,i+1:nrow);

b(j) = b(j) + m*b(i);

end

end

% Back substitution

x(nrow) = b(nrow) / A(nrow,nrow);

fprintf('\n\nHas exact solution:\n')

for i = nrow-1:-1:1

x(i) = (b(i) - sum(x(i+1:nrow) .* A(i,i+1:nrow))) / A(i,i);

end

end