(6) Let (G, *) be a group and a € G. Prove that the order of a is equal to the order of a-1. Prove also that the order of a is equal to the order of b* a*b-¹ for any b € G.

Q: Show that the group (R#, +) is isomorphic to the group (R#+, *), where R# is the set of real numbers…

A: R# is the set of real number and R#+ is the set of positive real numbers We have to show that the…

Q: 4. Suppose (G₁, *) and (G₂, 0) are groups, let i and J denote the respective identities of G₁ and…

A: and are group. be a group isomorphism.We have to prove that if  has finite order the  has finite…

Q: 3. In each part, find the order of the given element in the given group. (a) i= √-1 in the…

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Q: Show that conjugacy is an equivalence relation on a group.

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Q: Let G be a group with order 8. If a is an element of G such that the order of a is 4, find the order…

A: The order of an element a in a group G is the smallest positive integer n such that an=e, where e is…

Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: Let (G, *) and (H, *) be groups. The direct product of G and H is the set G × H equipped with…

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Q: Consider two groups, G 1 ​ and  G 2 ​ . Group G 1 ​ is defined as {1,a,a^2 ,a^3 } with operation ⋅ ⋅…

A: this basically states the following that any 2 cyclic groups of the same order are isomorphic.

Q: Q7/ Find all possible non-isomorphic groups of order 77.

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

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Q: Let G be a group, let f, g = G, and suppose that f~G 9. Show that the order of f is equal to the…

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Q: is a group with identity (eg, eH).

A:

Q: 4) Let set G be the Cartisian product of G1 × G2, where G1 and G2 are 2 groups. Define a binary…

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Q: 1. Let G and H be groups, and let g E G and he H. Find a formula for the order of the element (g, h)…

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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: Determine if the following sets with operations are groups: (a) R with the operation given by r* y =…

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