1971 AHSME Problem 6

Let \ast be the symbol denoting the binary operation on the set S of all non-zero real numbers as follows: For any two numbers a and b of S, a\ast b=2ab. Then the one of the following statements which is not true, is

\begin{array}{l} \textbf{(A) }\ast\text{ is commutative over }S \qquad \textbf{(B) }\ast\text{ is associative over }S\qquad \\ \textbf{(C) }\frac{1}{2}\text{ is an identity element for }\ast\text{ in }S\qquad \textbf{(D) }\text{Every element of }S\text{ has an inverse for }\ast\qquad \textbf{(E) }\dfrac{1}{2a}\text{ is an inverse for }\ast\text{ of the element }a\text{ of }S \end{array}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.