Finding order of the element

Question

Assignment:

Q1. List 4 integers in [2] as an element in Z₄
Q2. Using modular addition, circle+, compare your answers for Z₅: [3] + [4] and Z₆: [3] + [4].
Q3. Repeat the question above using modular multiplication, circle_dot.
Q4. Construct a Cayley table for the group Z₄ and, using the table, provide an analysis on how the table proves Z₄ is a group.
Q5. Prove the operation circle_dot on Z_n is commutative
Q6. Explain why the operation circle_dot is not a group in Z_n
Q7. Solve the equation x(1 3 2)=(1 3) in S₃
Q8. Find the order of the element (1 2)(3 4 5) in S₅
Q9. Determine the elements in each of the cyclic subgroups of Z₆. Give the order of each element in Z₆.
Q10. 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.
Q11. List the elements of Z₄ and find the order of Z₄. Find the order of one of the elements, [2]. Find [2]^-1.
Q12. 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.
Q13. For any integer n > 2, show that there are at least two elements in U(n) that satisfy x² = 1.
Q14. Prove that the set of of all rational numbers of the form 3^m6ⁿ, where m and n are integers, is a group under multiplication.
Q15. 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.
Q16. 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).
Q17. Find the order of the element (1 2)(3 4 5) in S₅.
Q18. What is the order of S₄ X S₄.
Q19. A finite group G has elements of orders p and q, where p and q are distinct primes. What can you conclude about |G|?
Q20. Prove that if p is a prime, then Z_p is an integral domain.

Provide complete and step by step solution for the question and show calculations and use formulas.