<s>[INST] Is $G = (\Bbb Q^*,a*b=\frac{a+b}{2})$ a group, monoid, semigroup or none of these? 
Is $G = (\Bbb Q^*,a*b=\frac{a+b}{2})$ a group, monoid, semigroup or none of these?

I tried to solve it by assuming $ a,b,c \in G $ such that $a*(b*c)=(a*b)*c$. Then$$\frac{a+\frac{b+c}{2}}{2} = \frac{\frac{a+b}{2}+c}{2}$$
and by taking $$ 2\times\left(  \frac{a+\frac{b+c}{2}}{2} =  \frac{\frac{a+b}{2}+c}{2} \right) $$
we have $$ a+\frac{b+c}{2} =\frac{a+b}{2}+c.$$
In my steps I have $(G,*)$ is not associative. Are my steps right?
 [/INST] Let $a=2$ and $b=c=1$. This should be a counterexample to associativity.
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