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 ∈ S. (2) Prove that (S, *) is a group under the above binary operation *. (3) Prove that (S, *) is abelian.

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 ∈ S. (2) Prove that (S, ) is a group under the above binary operation . (3) Prove that (S, *) is abelian.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.6: Quotient Groups

Problem 30E: Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.

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Question

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 ∈ S.
(2) Prove that (S, *) is a group under the above binary operation *.
(3) Prove that (S, *) is abelian.

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