I am studying Abstract Algebra in college and my exams are approaching fast.I need somebody to help me out to do a few exam papers.

I am going to post the questions below from the exam papers and if you can advise me how to do do them , just post what to do.I dont think that they are that difficult and if you have a good grasp of the algebra they are fairly doable.

Q1

State clearly the general definition of an equivalence relation. Show that
(a, b) ~(r, s) if and only if a + s = b + r
defines an equivalence relation on the set M = {1, 2, 3, 4, . . .}◊{1, 2, 3, 4, . . .},
which contains precisely all pairs of positive integers.


Q2

(b) State clearly Lagrangeís Theorem.
Find all subgroups of the symmetric group S3.

(c) Prove that {1, t, t2, t3, t4, t55 marks } is a normal subgroup of the dihedral group D6.

Q3
State Eulerís theorem and use it to compute 254477550 mod 282.

Q4
Prove that the multiplicative group (Z/24Z)* is not isomorphic to the additive group (Z/8Z).

Q5

Write  = (1 4 8 7)(3 4)(1 8 5) element of S8 as a product of cycles with disjoint trace.
Determine the order ord ( ) and parity sgn ( ) of the permutation Write its
inverse −1 as a product of cycles with disjoint trace.

All help would be appreciated. I dont think that they are too difficult but like everything in maths its just a matter of knowing how to do them..

PM me if necessary!