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 Lagrange’s 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 Euler’s 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!