You need to check if the 4 axioms hold such that:

1. Closure axiom: For all `g_1, g_2 in R =gt g_1og_2 in R`

Hence, for all `x,y in R` yields `xoy=x + y + sqrt3 ` `in R` (the addition of three real numbers, x+y+sqrt3, is also a real number)

2. Identity axiom: There is an identity element `e in R` such that:

`xoe = eox = x`

`xoe = x+e+sqrt3`

You need to set the equations `x+e+sqrt3` and x equals such that:

`x+e+sqrt3 = x =gt e+sqrt3 = 0 = gt e = -sqrt 3`

3. Inverses axiom: For each `x in R` , there is an inverse `x^(-1)``in R` such that:

`x o x^(-1) = x^(-1) o x = e`

`x o x^(-1) = x + x^(-1) + sqrt 3`

`x + x^(-1) + sqrt 3 = -sqrt 3 =gt x^(-1) = -x - 2sqrt3`

4. Associativity axiom: For all `x,y,z in R` yields:

`xo(yoz) = (xoy)oz`

`x + (yoz) + sqrt 3 = (xoy) + z + sqrt 3`

Reducing `sqrt 3` both sides yields:

`x + (yoz) = (xoy) + z`

You need to substitute `y+z+sqrt3` for (yoz) and `x+y+sqrt3 ` for (xoy) such that:

`x + y + z + sqrt3 = x + y + sqrt3 + z`

Notice that both sides are equal.

**Hence, since the 4 axioms hold, then the set R forms a group with binary operation "o".**

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