Let S = R\{-1} and define a binary operation on S by a * b = a +b+ ab. Prove that S is an abelian group. Note: you must prove that this structure has all of the properties of a group (associativity, identity, inverses), and also that the operation is commutative.

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

A: 2) S=R∖-1 binary operation defined by a*b=a+b+ab

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A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

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

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Q: Let S = R\ {−1} and define a binary operation on S by a * b = a+b+ab. (1) Show that a, b ∈ S, a * b…

A: Part A- Given: Let S=R\1 and define binary operation on S by  a*b=a+b+ab To show - a,b∈S,a*b∈S…

Q: A cyclic group is abelian

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Q: Prove that a group G is abelian if and only if (ab) = a¬!b-1 for all a and b in G.

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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.

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Q: Let (G, ) be a group. Define a new binary operation * on G by the formula a * b = b · a for all a, b…

A: We proved (G,*) is a group if it satisfied the following axioms.

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Q: Let (G, ) be a group. Define a new binary operation * on G by the formula a * b = b - a for all a…

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Q: Show that group U(1) is isomorphic to group SO(2)

A: See the attachment.

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A: See the attachment

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A: False Every element of cyclic group do not generate the group.

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