7. Let G be a group and for elements a, b e G define aRb if and only if there exists an element x E G such that a = xbx-1. Show that R is an equivalence relation on G.

Q: 5. E Prove that G is an abelian group if and only if the map given by f:G G, f(g) = g² is a…

A: The solution is given as

Q: 9.12 Let G be a group and for elements a, b e G let a R b mean that there exists an element x = G…

A:

Q: 5. Let G be the multiplicative group {[2], [4], [6], [8]} = Z10-

A:

Q: subgroups of G. Let x,y e G. We define the relation x ~ y by the following definition: x ~ y iff y =…

A:

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: 2. Are the groups (R, +) and (R†,') isomorphic? Justify your answer.

A:

Q: Suppose T: G-> H is group isomorphism and S:H-> K is also a group isomorphism. SHow the…

A:

Q: G be defined by f(r) = x1. Prove that f is operation-preserving if 6*. Let G be a group and f: G and…

A: To prove that the given function f is a homomorphism (operation preserving) if and only if G is…

Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

A:

Q: 1. Determine whether the following statements are true or false. Justify your answers. a. The set G…

A: a. Given set is G={0,1,2,3,4,5,6}. The set cannot be a group under multiplication modulo 7 as not…

Q: Let x and y be two vertices of a Cayley digraph. Explain why twopaths from x to y in the digraph…

A: x and y are two vertices of a Cayley digraph. ai’s and bj’s are generators of the Cayley digraph

Q: Let ℝ be the set of real numbers and let • denote the normal multiplication operation on ℝ. The…

A: Let ℝ be the set of real numbers and let • denote the normal multiplication operation on ℝ.

Q: Z, x is not a group.

A:

Q: Let the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: Let Zodd represent the set of odd integers, {... -5, -3, -1, 1, 3, 5, ...}. Determine if the system…

A: The given set is Zodd=. . . . . . -5, -3, -1, 1, 3, 5, . . . . . . . We have to determine  (i) Zodd,…

Q: 2 20. Let G be a group with subgroup H. Define a relation on G as follows: a b if b-¹a EH. Prove…

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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: (a) Consider (3,6) and (2,4) as elements of Z7xZ7. Compute (3,6) - (2,4). (b) Consider (3,6) and…

A: