In: Advanced Math

14. Extra Credit: Cayley’s Theorem is an important one in advanced algebra. It says that “Every algebraic group is isomorphic to some permutation group.” Demonstrate this to be true by finding a permutation group (Sn, ∘ ) that is isomorphic to (ℤ3, +) for some n.

we have verified cayleys theorem for the group .We have found an isomorphism to group of even permutations.so please see thenote above to get a better understanding.

Linear algebra Thank you.
Extra Credit (No partial credit - 1%): It is often useful to
perform a change of basis to examine an object from a different
perspective. In the process of a larger work the
conic work the conic 3 x square − 10 x y + 3 y square +
16 x root 2 − 32 = 0 needs to be identified; this can be
accomplished by a rotation. Identify the given conic and give its
equation in...

Central limit theorem is important because?
For a large , it says the population is approximately
normal.
For a large , it says the sampling distribution of the sample
mean is approximately normal, regardless of the shape of the
population.
For any population, it says the sampling distribution of the
sample mean is approximately normal, regardless of the sample
size.
For any sized sample, it says the sampling distribution of the
sample mean is approximately normal.

According to the Fundamental Theorem of Algebra, every
nonconstant polynomial f (x) ∈
C[x] with complex coefficients has a complex root.
(a) Prove every nonconstant polynomial with complex coefficients
is a product of linear polynomials.
(b) Use the result of the previous exercise to prove every
nonconstant polynomial with real coefficients is a product of
linear and quadratic polynomials with real coefficients.

Prove by induction that it follows from Fundamental Theorem of Algebra that every f(x) ∈ C[x] can be written into a product of linear polynomials in C[x].

Abstract Algebra and Galois Theory
Show how to one can deduce the Fundamental Theorem of Galois
theory from the Artin's lemma.

The Bible in Luke 14:28 says, “Suppose one of you wants to build
a tower. Won't you first sit down and estimate the cost to see if
you have enough money to complete it?” How do you think this verse
might apply to this chapter on closing entries? Where do you see
the connection between doing closing entries and whether a company
should proceed with a business venture?

derive the Pythagorean theorem (a2+b2=c2) which is one of the most important theorems in mathematics

Create 3 PowerShell Scripts. The first 2 are worth 2 points
each. The 3rd one is extra credit, worth 1 point.
Script 1: Add/Remove Users
Fields: Username, FirstName, LastName, Description, OUPath,
Password, Action (Add | Remove)
Script 2: Create Security Groups
Fields: Name, Description, OUPath, Scope (DomainLocal | Global |
Universal), Category (Distribution | Security)
Script 3: Modify Security Group Membership
Fields: GroupName, Username, Action (Add | Remove)

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