Problem 6 Assume x, y are elements of some group G, |x| = n and [y] = m. Suppose that x and y commute: xy = yx. 1. Prove that |xy| divides the least common multiple of m and n. 2. Give a counterexample to this statement if we don't assume that x and y commute. 3. Give an example of commuting elements x, y such that the order of xy is not equal to the least common multiple of |x| and [y].

Q: 4.12 write the group table for 2. U(40)/(3)

A:

Q: 1. Consider the group D5. Give five examples of D5 acting on a set, other than the trivial action.…

A: Given: Consider the group D5. Give five examples of D5 acting on a set, other than the trivial…

Q: 7. Consider an element z = the order of r is the least common multiple of the orders of its…

A:

Q: In (Z24, €) find the orders of the elements 3, 4, 6, 7, and 18.

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: 30. Prove the right and left cancellation laws for a group G; that is, show that in the group G, ba…

A:

Q: How would we show that H is cyclic?

A:

Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: Let Y = {2019, 2020} and M = {CRV, Accord, Civic} %3D List the elements of Y x M.

A: Given that Let Y ={2019,2020} and M = {CRV,Accord,Civic} List the elements of YxM Explanation :- YxM…

Q: 2. List all the elements of (Z30, Ð) that are of order 15.

A: