Abstract algebra proofs

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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 integral domain.