1. Consider the group D5. Give five examples of D5 acting on a set, other than the trivial action. For each one, explain clearly what the action is, although you do not need to prove that it is a group action.

Q: Evaluate if x=4,8,3,-2 ∑x ∑x+1 ∑(x+1) 3∑x - 4

A: Given that  if x=4,8,3,-2

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

A:

Q: In the frieze group F7, write x-3zxyz in the form xnymz k.

A: It is known that F7=xnymzk∥n∈Z,m=0,k=0 or k=1 and x is the translation of R2, and it map u,v to…

Q: Give an example: The product of two solvable groups need not to be solvable?

A: it is clear that s2 is solvable because it is abelian

Q: Define Group theory ?

A: To define group theory

Q: The group (Z4 ⨁ Z12)/<(2, 2)> is isomorphic to one of Z8, Z4 ⨁ Z2, orZ2 ⨁ Z2 ⨁ Z2. Determine…

A: Consider the group elements, Here the order of K is 6. Consider the order of group, The order of G…

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

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

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

A: In this question, we have to find the minimum number of generators needed for the group  ℤ2⊕ℤ2⊕ℤ2…

Q: This exercise refers to the triangular symmetry group discussed in this section.Which element is the…

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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: 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: In the frieze group F7, write x2yzxz in the form xnymz k.

A: It is given that, F7=xnymzk|n∈Z, m=0 or m=1, k=0 or k=1. we need to write x2yzxz  in the form of…

Q: For groups Gı and G2, determine the center of G1 × G2.

A:

Q: Let G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar groupis G isomorphic?

A: Here given that  G =<a, b | a2 =e, b2 =e, aba =bab>. To what familiar group is G isomorphic…

Q: Demonstrate or explain why the system (P2(R#), *) is NOT a semi-group, that is, demonstrate or…

A: To demonstrate or explain why: P2ℝ,* is NOT closed or P2ℝ,* is NOT associative

Q: Is (Zeven, +) a commutative group?  Explain

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Q: Show, by example, that in a factor group G/H it can happen thataH = bH but |a| ≠ |b|.

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Q: If X and Y are completely unrelated, r will be close to 1. Group of answer choices True False

A: If X and Y are completely unrelated, r will be close to 1. Group of answer choices True False

Q: Please explain an infinite p-group, and give an example

A: Infinite p-group: Infinite p-group is an infinite group in which the order of every element is a…

Q: For each of the following groups, find all the conjugacy classes and write out the conjugacy class…

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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: Q8: What is the difference between homomorphism and isomorphism?

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

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Q: Consider the groups  and .

A: We have to consider the groups

Q: (S) Is all groups of ve, glve aIl pie. about groups of order 5? (Are they always commutative).

A: Concept:

Q: Is it possible to find a group operation ⊕ on a set with 0 elements? With 1 element? Explain why or…

A: A finite or infinite set G is a group under the binary operation  if it satisfies the following four…

Q: What is a homomorphism in simplest terms? Also provide an example.

A:

Q: Show that isomophism between group is an equivalence relation. Briefly explain and show all the…

A: Recall the definition of an equivalence relation. It satisfies reflexive, symmetric and transitive.…

Q: Explain each of them with a easy understanding so people can understand fast and easy. Also prove…

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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: Why use f test when there are 4 groups? I would appreciate it if you could explain in detail.

A: Why use F test when there are 4 groups?  Concept Used: Independent t test is used for comparing one…