Abstract Algebra Questions:
1. List 4 integers in [2] as an element in Z4
2. Using modular addition, circle+, compare your answers for Z5: [3] + [4] and Z6: [3] + [4].
3. Repeat the question above using modular multiplication, circle_dot.
4. Construct a Cayley table for the group Z4 and, using the table, provide an analysis on how the table proves Z4 is a group.
5. Prove the operation circle_dot on Zn is commutative
6. Explain why the operation circle_dot is not a group in Zn
7. Solve the equation x(1 3 2)=(1 3) in S3
8. Find the order of the element (1 2)(3 4 5) in S5
9. Determine the elements in each of the cyclic subgroups of Z6. Give the order of each element in Z6.
10. Let (a b c) → (a b c) be a mapping such that a*b → b. Develop a Cayley table using the operation * and the mapping just described. There is only one way to form a group. Find it.
11. List the elements of Z4 and find the order of Z4. Find the order of one of the elements, [2]. Find [2]-1.
12. List the elements of S3 and find the order of S3. Find the order of one of the elements, (1 3 2). Find (1 3 2)-1.
13. For any integer n > 2, show that there are at least two elements in U(n) that satisfy x2 = 1.
14. Prove that the set of of all rational numbers of the form 3m6n, where m and n are integers, is a group under multiplication.
15. Let G be the group of nonzero real numbers under multiplication. H = {x ∈ | x = 1 or x is irrational} and K = {x ∈ G | x ≥ 1}. Provide a proof for both H & K that they are or are not a subgroup.
16. The subgroup (slightly different symbology is used in your text) is called the cyclic subgroup of a group G generated by a. If G = , we say G is cyclic and a is a generator of G. For example, in Z10 <2> = {2,4,6,8,0} since an = na when the operation is addition. Show that in U(10), <3> = U(10); that <3> is a generator of U(10).
17. Find the order of the element (1 2)(3 4 5) in S5.
18. What is the order of S4 X S4.
19. A finite group G has elements of orders p and q, where p and q are distinct primes. What can you conclude about |G|?
20. Prove that if p is a prime, then Zp is an integraldomain.

Hi! I'm using old tests to study for an abstractalgebra exam and of course the old exams do not come with solutions. This means that I need help. I tend to struggle with proofs because I forget some steps or I am not as rigorous as I should be. Thus I need to see actual complete rigorous proofs so that I can make sure that I re

Chapter 0 (Preliminaries)
Q.) How would you prove the converse?
A partition of a set S defines on equivalence relation on S.
Hint: Define a relation as X - Y if X and Y are elements of the same subset of the partition.
10.) Let n be fixed positive integer greater than 1. If a mod n = a' and b mod n = b' .Prove that (a

Given:
Define a geometric construction as an object that can be created using only a compass and a straightedge. Mathematicians have shown that it is not possible to:
1. Geometrically construct a square with an area equal to that of a given circle.
2. Use a geometric construction to trisect an arbitrary angle.
The proo

1. Define (C_G)(H) = {g is a number in G: g h = h g for all h is a number in H), where H is a subgroup of the group G. Prove that (C_G)(H) is a subgroup of G. Note: (C_G)(H) is called the centralizer of H in G.
2. Define (N_G)(H) = {g is a number in G: gH = Hg], where H is a subgroup of the Group G. Prove that (N_G)(H) is a s

Problem 1: Given the metric space (X, p), prove that
a) |p(x, z) - p(y, u)| < p(x, y) + p(z, u) (x, y, z, u is an element of X);
b) |p(x, y) - p(y, z)| < p(x, y) (x, y, z is an element of X).
These problems are from Metric Space. Please give formal proofs for both (a) and (b) based on the reference provided. Thank y

Please help with the following proofs. Answer true or false for each along with step by step proofs.
1) Prove that all integers a,b,p, with p>0 and q>0 that
((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q
Or give a counterexample
2) prove for all integers a,b,p,q with p>0 and q>0 that
((a-b)mod p) mod q=0

See the attachment for all questions.
18. Given: If you are wealthy, then you are a success.
You are wealthy or you are. healthy.
You are not healthy.
Let W represent: "You are wealthy." Let S represent: "You are a success." Let H represent: "You are healthy."
Prove: You are a success.
19. Given: The object i