The set $\{-1, 0, 1\}$ is not a multiplicative group because of the failure of

Show that $-a$ is not the inverse of $a \in N$ for the addition operation $+$ on $N$ and $\frac{1}{a}$ is not the inverse of $a \in N$ for the multiplication operation $\times$ on $N$,for $a \neq 1$.

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

If $a * b = 10ab$ on $Q^{+}$,then find the inverse of $0.01$.

Show that addition, subtraction, and multiplication are binary operations on $R$, but division is not a binary operation on $R$. Further, show that division is a binary operation on the set $R_*$ of nonzero real numbers.

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