Determine whether the binary operation * gives a group structure on the given set. If no group results, give the first axiom in the order G₁, G₂, G₃ from Definition 4.1 that does not hold.

Let* be defined on Q by letting a* b = ab.

Data from Definition 4.1.

A group (G, *) is a set G, closed under a binary operation *, such that the following axioms are satisfied: 

G₁: For all a, b, c ∈ G, we have 

(a * b) * c =a* (b * c). associativity of *

G₂: There is an element e in G such that for all x ∈ G, 

e * x = x * e = x. identity element e for* 

G₃: Corresponding to each a ∈ G, there is an element a' in G such that 

a* a' =a'* a = e. inverse a' of a