Let $G = \mathbb R\setminus\{-1\}$. Prove that G is a group with respect to the binary operation $x*y = xy + x + y$, where $xy$ and $x + y$ denote the usual multiplication and addition of reals, respectively.
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1Just check the group axioms. I don't think it is something hard. – Mark Mar 28 '19 at 19:29
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What have you tried so far ? You just have to verify the group axioms, i.e. that $G$ is closed under $$; that it contains inverses w.r.t. to $$; that it contains a unit for $$ and that $$ is associative. – blub Mar 28 '19 at 19:30
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Note that $xy+x+y=(x+1)(y+1)-1$. As such, since the reals are an integral domain, the only way $x*y=-1$ is if $x+1=0$ or $y+1=0$. That shows that the set is closed under the operation. The rest of the properties hold from the corresponding properties of multiplication and addition. – Rushabh Mehta Mar 28 '19 at 19:32
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See also this question. – Dietrich Burde Mar 28 '19 at 19:35