Questions
In a finite cyclic group, each subgroup has size dividing the size of the group. Conversely, given a positive divisor of the size of the group, there is a subgroup of that size

Prove that in a finite cyclic group, each subgroup has size dividing the size of the group. Conversely, given a positive divisor of the size of the group, there is a subgroup of that size.

In: Advanced Math

Measure Theory

Let f be measurable and B a Borel set. Then f-1[B] is a measurable set. [Hint: The class of sets for which f-1[E] is measurable is a σ-algebra.

In: Advanced Math

Let G be finite with |G| > 1. If Aut(G) acts transitively on G − {e} then G ∼= (Z/(p))n for some prime p.

Let G be finite with |G| > 1. If Aut(G) acts transitively on G − {e} then G ∼= (Z/(p))n for some prime p.

In: Advanced Math

How do you calculate the probability of compound events?

How do you calculate the probability of compound events?

In: Advanced Math

Find and Simplify where ................................

Find f(a+h) - f(a) / h where f(x) = 6x - 9

In: Advanced Math

What is the number of group homomorphisms from z12 to z13?

What is the number of group homomorphisms from z12 to z13?

In: Advanced Math

What is the vertex of y = 2x² + 6 x + 4 ?

What is the vertex of  

y=2x²+6x+4  ? Give details explaination

In: Advanced Math

concave function

What x values is the function concave down if  f(x)=15x⅔+5x ?

In: Advanced Math

Number Theory

Given integers a, b, c,

g.c.d.(a, b, c) = 1 if and only if g.c.d.(a, b) = 1 and g.c.d.(a, c) = 1

In: Advanced Math

syllw subgroup

The number of involutions in G is |G|/4, and every right coset of a Sylow 2-subgroup S of G not contained in NG(S) contains exactly one involution.

In: Advanced Math

Integer property ( Real Analysis)

An integer n is called even if n = 2m for some integer m, and odd if n + 1 is even. Prove the following statements:

(a) An integer cannot be both even and odd.

(b) Every integer is either even or odd.

(c) The sum or product of even integers is an even integer. What can you say about the sum or product of odd integers?

In: Advanced Math

Density property of Real number

If x and y are arbitrary real numbers such that x < y, prove that there exists at least one rational number r satisfying x < r < y, and hence infinitely many.

In: Advanced Math

Differential equation

Solve the differential equation with details explaination :

x²y" + 6xy' - 24y=x^9

In: Advanced Math

Kernal ( Homomorphisms)

Prove that φ : Z ⊕ Z → Z by φ(a, b) = a − b is a homomorphism. Determine the kernel.

In: Advanced Math

Homomorphisms

How many (group) homomorphisms are there from Z20 onto (surjective to) Z8. How many are there to Z8?

In: Advanced Math