(a) G = Z where a * b = a + b – 1 for all a, b EG (b) G = {r € Q |r > 0 } where a * b = ab for all a, b e G (c) G = R\{1} where a * b = a – ab + b for all a, b e G

Q: 1 10. Given a 3 x 3 matrix A that has its inverse matrix A- = 3 4 -2 -6 -1 1 determine the value of…

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Q: Verify that (Z, O) is an infinite group, where Z is the set of integers and the binary operator O is…

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Q: Let m ≥ 3. Prove that the symmetric group on m letters is not abelian.

A: We have to show that the permutation of m letters, m≥3 that is Sm is not abelian. For proving the…

Q: Let a be a group element of even order. Prove that a2 is not in cl(a).

A: Let G be a group and a is an element of even order in the group. Prove that a2 is not in Cla.…

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Q: Let G be an Abelian group. Determine all homomorphisms fromS3 to G.

A: Given: G is an Abelian group.To find: All homomorphisms from S3→G.

Q: b. Find all abelian groups, up to isomorphism, of order 360.

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Q: (d) Define * on Q by a * b = ab. Determine whether the binary operation * gives a group on a given…

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Q: ose a Let X be a finite set with at least two elements. Suppose group Gr acti transitively on X.…

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Q: Assume that G is an abelian group. Let H C G be a subgroup. Prove that the factor group G/H is…

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Q: If G is a group, prove that Aut(G) and Inn(G) are groups

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Q: For any group G then G/Z(G) is abelin Select one: O True O False

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Q: Show that (up to isomorphism) A5 is the only simple group of order60.

A: Given : G=60 To Prove : G≃A5

Q: Let G be a group (not ncesssarily an Abelian group) of order 425. Prove that G must have an element…

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Q: Prove that in a group, (a-1)¯' = a for all a.

A: To prove that in a group (a-1 )-1=a for all a.

Q: Any product of solvable groups is solvable.

A: Prove that, any product of solvable group is solvable.

Q: Every cyclic group is necessarily abelian but the converse is not necessarily true

A: Given statement :Every cyclic group is necessarily abelian but the converse is not necessarily true…

Q: Let S = R\{-1} and define a binary operation on S by a * b = a +b+ ab. Prove that S is an abelian…

A: To show that S is an abelian group, we have to prove all these properties 1) S is closed under…

Q: Prove that An even permutation is group w.y.t compostin Compostin function.

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Q: Every cyclic group is necessarily abelian but the  converse is not necessarily true.

A: Given statement :Every cyclic group is necessarily abelian but the converse is not necessarily true…

Q: Let G be a group, and a, b € G. Prove that b commutes with a if and only if b- commutes with a.

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Q: Let X be any non-empty set. Show that the set of all bijections of X is a group under the operation…

A: Given: X is any non-empty set.To show: The set of all bijections of X is a group under the operation…

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

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Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Q)Let G be a group such thatx=x- for each xeG. Show that G is Abelian

A: Given : G is a group such that x=x-1 for each x∈G   To prove : G is abelian.