Let 𝑄∗ be the set of non-zero rational numbers and * be the binary operation on Q definedby 𝑎 ∗ 𝑏 = 𝑎𝑏/3. Prove that 𝑄∗ is group under *.

To prove that Q* is a group under *, we need to show that it satisfies the four group axioms: closure, associativity, identity, and invertibility.

1. Closure: For any two elements a, b in Q*, ab = ab/3 is also in Q because the product and division of non-zero rational numbers is also a non-zero rational number.

2. Associativity: For any three elements a, b, c in Q*, we need to show that (ab)c = a(bc). (ab)c = (ab/3)c = abc/3^2 a(bc) = a(bc/3) = abc/3^2 Therefore, (ab)c = a(bc), so * is associative.

3. Identity: We need to find an element e in Q* such that for every element a in Q*, ae = ea = a. Let's try e = 3. a3 = a3/3 = a and 3a = 3a/3 = a. So, e = 3 is the identity element.

4. Invertibility: For each element a in Q*, there must exist an element b in Q* such that ab = ba = e. Let's try b = 1/a. Then ab = a(1/a) = a/a = 1 and b*a = (1/a)*a = a/a = 1. So, b = 1/a is the inverse of a.

Therefore, Q* is a group under *.