In a group $(G, *)$,for some element $a$ of $G$,if $a^{2}=e$,where $e$ is the identity element,then

In the group $(G, \times_{15})$,where $G = \{3, 6, 9, 12\}$ and $\times_{15}$ is multiplication modulo $15$,the identity element is

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 $a$.

For each binary operation $^*$ defined below,determine whether $^*$ is commutative or associative. On $Q$,define $a ^* b = ab + 1$.

In the set of integers $(Z, *)$,if $a * b = a + b - n, \forall a, b \in Z$,where $n$ is a fixed integer,then the inverse of $(-n)$ is:

If $A = \{a, b, c\}$,then the number of binary operations on $A$ is