Abstract Algebra Questions - Need help for exam

[goldstar]

Abstract Algebra Questions - Need help for exam!

Homework Statement

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 don't 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) iff 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 don't think that they are too difficult but like everything in maths its just a matter of knowing how to do them..

 

[Dick]

I would suggest you pick one of those problems, make an honest attempt to solve it and then post where you are having problems. Follow the submission form posting all needed definitions of the symbols you are using. I don't even know what some of those questions mean.

 

[HallsofIvy]

The first thing you need to do is look up the DEFINITIONS of the terms you are using. The first problem, for example, says "state the definition and the second part of it follows immediately from the definition of "equivalence relation". If you know what "Lagrange’s Theorem", "dihedral group", "Euler's theorem" etc. are, yes, the problems are easy.

 

[goldstar]

Ok i will look them up. I am just up the walls at the moment with assignments and study. If you could explain just how to do these few questions i would be very grateful .I don't think that they are very difficult.

Langranges theorem refers to cosets and groups and the number of each which you can have. I have the definition of an equivalace relation also.

These problems are not too difficult but would be tricky enough for me to solve ,please give me a start on them as I can use them as the basis to solve other problems...

Q4 ,Q2 b and c are the ones I am finding most trouble with.

 

[HallsofIvy]

Saying "Lagranges theorem refers to cosets and groups and the number of each which you can have" is not enough. Exactly what does Lagranges theorem say?

I'm glad you have the definition of an equivalence relation. Now it should be obvious that to prove something is an equivalence relation, you only have to show that the conditions given in the definition are true for this particular relation.
What are those conditions?