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: 54 The groups An and Sn are not solvable for all n ≥ 5.

A: Solution:-

Q: c. Complete the following statements: The group Sg = Sym (Ng) of all permutations on the set Ng…

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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: What is the order of the symmetry group in R3 of a 4-prism (a boxwith square ends that is not a…

A: Given: Symmetry group in R3 of a 4-prism.To find: The order of given symmetry group.

Q: What is the order of the (entire) symmetry group in R3of a 3-prism?

A: A 3-prism is a polyhedron with two congruent and parallel faces (called the bases) and rectangular…

Q: X={1,2,3,4} .The permutation group S4=

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Q: In practice, what can be the differences between the relationships of two pairs of groups, knowing…

A: Homomorphism -  It is  algebraic term for a function having some algebraic operations. For a group…

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: How many groups are there of order 12 (up to isomorphism) 5 O None of them 2 O 4 O 8 0

A: Use the concept of isomorphism.

Q: The product of 16 and 7.5 belongs to what group in the Real Number System?

A: Assuming the product means multiplication. (R\{0},*) forms group under multiplication. Also,…

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: Suppose that " is in lowest terms and that m< n. Then has a simple-periodic dec- imal representation…

A: Introduction: The ratio of given two numbers could be represented as a factor less than one always…

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: 4. Let G = {a+b√2, a, b E Z+}, * is ordinary multiplication. Is G a group?

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Q: If pq and r are three distinct primes and G is a group of order par, then G is not simple. .

A: Suppose that p,q and r are three distinct primes and G is a group of order pqr, then prove that G is…

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…

Q: 1. Define the multiplication operation as modulo 10 in G = {2,4,6.8}. Determine whether the system…

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