Let G = {q ∈Q / q ≠ -1}. Define the binary operation o on G by
x o y = x + y + xy. Prove that (G, o) is an abelian group.

To prove that \((G, o)\) is an abelian group, we need to verify that the set \(G = \{ q \in \mathbb{Q} \mid q \neq -1 \}\) with the binary operation \(o\) defined by \(x \, o \, y = x + y + xy\) satisfies the four properties of an abelian group:

1. **Closure**: \(G\) is closed under the operation \(o\).
2. **Associativity**: The operation \(o\) is associative.
3. **Identity element**: There exists an identity element \(e \in G\) such that \(x \, o \, e = e \, o \, x = x\) for all \(x \in G\).
4. **Inverse element**: For each element \(x

Let G = {q ∈ Q / q ≠ -1}. Define the binary operation o on G by x o y = x + y + xy. Prove that (G, o) is an abelian group.

In this detailed solution, we prove that the set G = {q ∈ Q / q ≠ -1} with the binary operation o defined as x o y = x + y + xy forms an abelian group. Learn about the key concepts of group theory, including closure, associativity, identity, inverse, and commutativity.

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