If θ: <R*, x> ---> <R*,x> is homomorphism such that θ(x) = x^3.

My first question:
if θ: ---> is homomorphism such that θ(x) = x^3. What is the size of the kernel of θ
Please show me the steps to get the answer. I try and my answer is 3.

My 2nd question: This i dont understand how to begin
If θ:(Q,*) ---> (Q,*x) given by θ(x) = 3x-1 is a group homomorphism, where the binary operation in (Q,*) is a*b = 3ab - a - b +2/3. What is the identity element e of (Q, *)?

Thank you in advance

1) ker θ = {x in R* | θ(x) = 1}
...........= {x in R* | x^3 = 1}
...........= {1}, since the other two roots are not real.

Hence, |ker θ| = 1.
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2) Under a homomorphism, the identity element of the first group is mapped to the identity of the second group. So, the identity e of (Q, *) satisfies θ(x) = 3x - 1 = 1 ==> x = 2/3.

Check: For all a in (Q, *),
a * 2/3 = 3(a)(2/3) - a - 2/3 + 2/3 = a, as required. (Similarly, 2/3 * a = a.)
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I hope this helps!