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Hi I'm stuck on this homework question: "For a,b∈R we define a*b:=a+b+ab∈R. Furthermore let G:=R/{−1}.

Show that G together with the binary operation G × G → G, (a, b) → a*b, is a group."

So I know I need to show that it's associative, there exists a neutral element and there exists an inverse element. What I'm stuck on is what is a*(bc) cause I know bc=b+c+bc so what's a*(b+c+bc). Any help would be very much appreciated.

Thomas
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    Probable duplicate of http://math.stackexchange.com/questions/513045/need-to-prove-that-s-defined-by-the-binary-operation-ab-abab-is-an-abel/ – lhf Feb 08 '17 at 11:25

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Just keep plugging in. We have $$ \begin{align}a*(b*c)&=a*(b+c+bc)\\&=a+(b+c+bc)+a(b+c+bc)\\&=a+b+c\;+\;ab+ac+bc\;+\;abc\end{align}$$

Hagen von Eitzen
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The map $f(x)=1+x$ maps $G$ bijectively to $\mathbb R^*$ and the operation on $G$ corresponds to ordinary multiplication in $\mathbb R^*$:

$$a * b = f^{-1}(f(a)\cdot f(b))$$

We then say that $G$ is a group by pullback or transport of structure from the group structure in $\mathbb R^*$.

$f$ is then by construction an isomorphism.

lhf
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