Abstract Algebra Answers

P rove that all real of the form a + b 2 , a; b ∈ Z form s a ring.

Let G be a group of order 11 2 :13 2 how many 11sylow subgroups and 13 sylow subgroups are there in G

 Let A = {1,2,3,4,5,6} and let p1 = (3,6,2) and p2 = (5, 1, 4) be permutations of A.

(a) Compute p1 ◦ p2 and write the result as a product of cycles and as the product of transpositions.

(b) Compute p−1 ◦ p−1. 

Prove that all real of the form  a + b 2, a; b ∈ Z forms a ring

Find aba-1 where (i) a = (5, 7, 9) , b = (1, 2, 3) (ii) a = (1, 2,5)(3, 4) , b = (1, 4, 5) .

If G is the abelian group of integers in the mapping T: G → G given by T(x ) = x then prove that as an automorphism

Let G be a group of order 112:132 . How many 11-sylow subgroups and 13 sylow subgroups are there in G?