Abstract Algebra Answers

Express the following as a product of disjoint cycle. Check them for being even or odd permutation and find the inverse of each of them in S7

1. (1 4 7) (2 6 5) (2 4 1) (5 6 7)

2. (7 1) (2 6 5) (2 4 1) (7 5 6)

Express the following permutation in S7 in two line notation

(1 4 7) ( 2 6 5)

If R is noetherian and I is a ideal show that R/I is noetherian.

Direct sum of neotherian module's is neotherian

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Express following permutation in S6 in cyclic notation a=(1 2 3 4 5 6

3 4 1 5 6 2)

In the set R of real numbers define an algebraic operation α by

α(a,b)=a-b

(a) (R,α) is not group

(b) α is not commutative

Let A={a,b,c} and P(A) the power set of A. List all the element of P(A). Show that the usual intersection, ∩ , and Union, U, of sets in P(A) are algebraic operation. What are the cayley's tables for there operation, Find the identity element if any, with respect to these operation.

Let π(N) be the number of primes less than or equal to N (example: π(100) = 25). The famous prime number theorem then states (with ∼ meaning asymptotically equal):
π(N) ∼ N/ log(N)
Proving this theorem is very hard. However, we can derive a statistical form of the prime number theorem. For this, we consider random primes which are generated as follows:
(i) Create a list of consecutive integers from 2 to N.