Which of the following is not a group with respect to the given operation?

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = \frac{ab}{2}$.

If $*$ is a binary operation defined on $R$ by $a * b = 1 + ab, \forall a, b \in R$. Then the operation $*$ is:

Let $*$ be a binary operation on the set $Q$ of rational numbers defined as $a * b = \frac{ab}{4}$. Which of the following is true?

Define a binary operation $^*$ on the set $\{0, 1, 2, 3, 4, 5\}$ as $a \,^* \, b = \begin{cases} a+b, & \text{If } a+b < 6 \\ a+b-6, & \text{If } a+b \geq 6 \end{cases}$. Show that $0$ is the identity for this operation and each element $a \neq 0$ of the set is invertible with $6-a$ being the inverse of $a$.

Let $*$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a * b = a^{2} + b^{2}$ for all $a, b \in Q$ is commutative.