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Let P be a normal Sylow p-subgroup of G and let H be any subgroup of G. Prove that P intersect H is the unique Sylow p-subgroup of H.
Show that a group G of order 2p^n has proper normal subgroup, where p is odd prime number and n > 0.
Using examples and/or theorems, argue that G has at least one subgroup of every order dividing |G|.
Let G be a group with A and B subgroups of G. Prove that the set AB = {ab | a is in A and b is in B} is a subgroup of G if and only if AB = BA
Let G be a group and S any subset of G. Prove that C_G (S) = {g in G such that gs = sg for all s in S} is a subgroup of G.
G1=G2=Z5 (the integers modulo 5 with the operation +), f:n?an(mod5) where a?Z_5\{0}={1,2,3,4} is fixed.
Give two reasons why the set of odd integers would not form a group under ordinary addition.
When an element and its inverse are combined under and operation the result is the identity element. The identity element for multiplication is 1.
Let i be an integer with 1 <= i <= n. Let Gi* be the subset of G1 X ... X Gn consisting of those elements whose ith coorinate is any element of Gi.
Each one need only be 1 to just a few sentences - please list separately and under each fact provide any links to websites or pictures etc.
Find all the integers such that when the final digit is deleted the new integer divides the original one.
A = U X U, where U = {1, 2, 3, 4, 6, 12}, and %:A -> Q is defined by %(n, m) = n/m, where Q is the set of all rational numbers.
Let G = GL(2,R) be the general linear group. Let H=GL(2,Q) and K= SL(2,R) = {A is an element of G: det (A) =1} Show that H,K are subgroups of G.
When the number is divided by 9 the remainder is 7 and when it is divided by 5, the remainder is 1. What is A-B=?
How much money has Lose-a-digit budgeted for the year? "The amount of the budget just happens to be the smallest number of cents (other than one cent).
Let I be an open interval containing the point x.(x not), and suppose that the function f:I->R has a continuous third derivative with f'''(x)>0 for all x in I.
Given ( INTEGRAL ln square(x)dx, as x from n to n+1 ) = ( INTEGRAL ln square (n+x)dx, as x from 0 to 1 ) = ( INTEGRAL [[ln(n+x) - ln(x) + ln(n)]square]
Let f(x) and g(x) be nonzero polynomials in R[x] and assume that the leading coefficient of one of them is a unit.
Write a polynomial that represents the total cost of Materilas and Labor for producing x transmissions.
A rectangular parking lot is 50 ft longer than it is wide. Determine the dimensions of the parking lot if it measures 250 ft diagonally.
When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative.
Determining a solution (an ordered pair) from a system of equations.
Linear Algebra - Vector Spaces.Let P be the set of all polynomials. Show that P, with the usual addition and scalar multiplication of functions
Determine whether the given sets form subspaces of R2. What does the T stand for in these equations?