(i) Closure axiom : 

Let x, y ∈ G. Then x = a + b√2, y = c + d√2; a, b, c, d ∈ Q. 

x + y = (a + b√2) + (c + d√2) = (a + c) + (b + d) √2 ∈ G, 

Since (a + c) and (b + d) are rational numbers. 

∴ G is closed with respect to addition.

(ii) Associative axiom : Since the elements of G are all real numbers, addition is associative.

(iii) Identity axiom : There exists 0 = 0 + 0√2 ∈ G such that for all x = a + b√2 ∈ G. 

x + 0 = (a + b√2) + (0 + 0√2) 

= a + b√2 = x 

Similarly, we have 0 + x = x. 

∴ 0 is the identity element of G and satisfies the identity axiom.

(iv) Inverse axiom: For each x = a + b√2 ∈ G, there exists -x = (-a) + (-b) √2 ∈ G such that x + (-x) = (a + b√2) + ((-a) + (- b) √2) 

= (a + (-a)) + (b + (-b)) √2 = 0 

Similarly, we have (- x) + x = 0 . 

∴ (- a) + (-b) √2 is the inverse of a + b√2 and satisfies the inverse axiom.

(v) Commutative axiom: 

x + y = (a + c) + (b + d) √2 = (c + a) + (d + b) √2

= (c + d√2) + (a + b√2) 

= y + x, for all x, y ∈ G. 

∴ The commutative property is true. 

∴ (G, +) is an abelian group. Since G is infinite, we see that (G, +) is an infinite abelian group.