1. a * b = 2ab = 2ba = b * a
Therefore commutative
a * (b * c) = a * (2bc) = 4abc
(a * b) * c = (2ab) * c = 4 abc
Therefore is associative.
2. a * e = a ⇒ 2ae = a ⇒ e = \(\frac{1}{2}\)
e * a = a ⇒ 2ea = a ⇒ e = \(\frac{1}{2}\)
Therefore identity element is \(\frac{1}{2}\).
3. a * b = \(\frac{1}{2}\) ⇒ 2ab = \(\frac{1}{2}\) ⇒ b = \(\frac{1}{4}\) a, a ≠ 0.