2. a) The operation in a finite group can be specified by writing down a table. Write the table for (Z9,) and use it to show that (Z9,) is a group. Find the inverse of 2 and 5 in (Z9,0).

Q: THE SET (1, 2, 4, 7, 8, 11, 13, 14) IS A GROUP UNDER MULTIPLICATION MODULO 15. THE INVERSES OF 4 AND…

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Q: What is the minimum number of generators needed for Z2 ⨁ Z2 ⨁ Z2?Find a set of generators and…

A: Given: Group, Z2⊕Z2⊕Z2To find: The minimum number of generators and a set of generators and…

Q: Compute the factor group (Z6 x Z4)/(([2]6, [2]4))

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Q: When do you use a t-test for independent groups and what do the numerator and deniminator in the…

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Q: Build the operation table ∗ on G = {e, a, b} knowing that (G, ∗) is a group.

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Q: 24. If a and b are distinct group elements, prove that either a² + b² or a³ + b³.

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Q: Give an example of a group of order 12 that has more than one subgroupof order 6.

A: Consider the group as follows, The order of a group is,

Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

Q: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: Show that (2*,A) is an abelian group. Make sure that you prove associativity, that you identify the…

A: First to prove that the given group is abelian. Here in the given group G is the set of all subsets…

Q: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.

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Q: Give an example of a p-group of order 9.

A: Given, Give an example of a p-group of order 9.

Q: For each integer n > 1, give examples of two nonisomorphic groupsof order n2.

A: To Do: For each integer n > 1, give examples of two non-isomorphic groups of order n2.

Q: How many elements of order 5 might be contained in a group of order 20?

A: using  third Sylow Theorem

Q: Let Zeven represent the set of even integers, {... -6, -4, -2, 0, 2, 4, 6, ...}. Determine if the…

A: (i) Closed: Yes, the set of even integers is closed under addition, meaning that when two even…

Q: The inverse of each element in the group ({1, 3, 5, 7}, ∙8)

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Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

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Q: 4. Consider the group A4, a group with twelve elements. Note that the positive divisors of 12 are 2,…

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Q: Show that Z2 is the only group that has exactly two conjugacy classes.

A: Given a group Z2.We have to show that, Z2 is the only group that has exactly two conjugacy classes.

Q: write why, in order to describe the homomorphic images of a group, it suffices to describe the…

A: The homomorphic images of a group refer to the set of all possible images that result from applying…

Q: The number of generators of a cyclic group of order 213 is * 48 24 144 140

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Q: Give an example, with justification, of a semi-group which is not a group.

A: A semigroup defined on a set is a binary operator which satisfies the association property of…

Q: Q2: If G = R- {0} and a * b = 4ab ,show that (G,*) forms a commutative group? %3D

A: To show for the commutative group of (G, *), we verify the following properties of the commutative…

Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

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Q: Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same…

A: To investigate the orders of the elements in the given groups

Q: 29. Let a and b be elements of a finite group G. a. Prove that a and a¯' have the same order. b.…

A: Given that a and b be elements of a finite group G.

Q: Find the inverse of the element 9 in the group (U(10), ®10).

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Q: Give an example of a monoid which is not group. Give an example of a groupoid which is not a…

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Q: Q2 / If a, b and c are elements of a group (G,*) such that c*a=c*b then a=b.

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Q: Find S4/N elements and show that they can be combined into a group. N={e,(23),(234),(243),(12)(34),…

A: We know,      | S4 | = 4! = 24     | N | = 12   So, | S4/N | = | S4 |/| N |                      =…

Q: Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.

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Q: The set {0, 3, 6, 9, 12, 15, 18, 21} is a group under the binary operation +24. Determine all its…

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Q: 3. consider the group D4 = {(1), (1234), (13), (24), (1432), (12), (34), (14), (23), (13), (24)} a.…

A: To enumerate all the distinct subgroups of the group D₄, we need to consider all possible…

Q: 46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?

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Q: How to complete the table using the binary operation as a group? Let, S={f,g,h,j}, and let the…

A: Given the set