Which of the following are groups? For those which are not groups, explain why not (it is enough to find one axiom that fails). For those which are, state the identity and inverses: Q under ×, {q∈Q:q>0} under ×, {q ∈ Q : q > 0} under division,  {1,2,3,4,5,6,7} under × mod 8, {1,3,5,7} under × mod 8, {2, 4, 6, 8, 10, 12} under × mod 14, The set {0,1,2,3,4,5} under the operation x ◦ y = |x + y − 5|.

Q: If n is not prime, then G = {1, 2, 3,..., n-1} is not a group under multiplication mod n.

A: We prove this result by contradiction.  Let n is not prime and G={1,2,3,....,n-1} is a group under…

Q: Is the set of integers a commutative group under the operation of addition? Yes; it satisfies the…

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Q: If p is a prime number the set {1,2,3,...,(p-1)}  is a group under multiplication modulo p.

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Q: 4. A binary operation a*b = 1/(ab) defined on positive Rational Numbers (Q+) is A. just a groupoid…

A: Use the relevant definitions of group theory.

Q: Question 3 [6 points each]. What is the inverse of each of the following elements: 1. 3 in (Z100,…

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Q: For each divisor k>1 of n, let U,(n) = {x E U(n) | x mod k = 1}. [For example, U,(21) = {1,4, 10,…

A: For each divisor k > 1 of n , Uk(n) =x∈U(n)| xmod k =1. To Find: List the elements of U420,U520,…

Q: Verify that each of the sets {1,3,7,9,11,13,17,19}, {1,3,7,9}, {1,9,13,17} forms a group under…

A: We can verify if each set forms a group under multiplication modulo 20 by checking the following…

Q: Suppose chips are randomly drawn from a bag, and each chip is labeled with either 1, 2, 3, 4, 5, 6,…

A: S={1, 2, 3, 4, 5, 6, 7, 8, 9} n(S) = 9 E = odd number = {1,3,5,7,9} n(E) = 5

Q: Suppose that you have $11,000 in a rather risky investment recommended by your financial advisor.…

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Q: "Proof": Take any group of n cats, where n = 1, 2, ... We need to prove that all cats in this group…

A: We need to find the error in the given induction proof.Let  denote the statement that the group of…

Q: The region bounded by f(x) 2x² + 12x + 14, x = = 0 is rotated about the y-axis. Find the volume of…

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Q: 10. If 4,42..a, is an element in a group, what is (4,az..a,) = ?

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Q: Prove that if n is not prime, then {1, 2, 3,..., n-1} is not a group under multiplication mod n.

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Q: How would we show that H is cyclic?

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Q: 1. 3 in (Z90, +90) 2. 4 in U(21), the group of units modulo 21.

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Q: 14. A number of grade nine students were surveyed and asked the question "Do you Feel Safe at…

A: Total number of opinions in the survey=6+20+35+29=90 Category Number of opinions corresponding to…