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

Let $^*$ be the binary operation on $N$ given by $a ^* b = \text{L.C.M. of } a \text{ and } b$. Find which elements of $N$ are invertible for the operation $^*$?

Let $^*$ be the binary operation on $N$ defined by $a \,^*\, b = \text{H.C.F. of } a \text{ and } b$. Is $^*$ commutative? Is $^*$ associative? Does there exist an identity for this binary operation on $N$?

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

Let $*$ be a binary operation defined on $R$ by $a * b = \frac{a+b}{4}$ for all $a, b \in R$. Then the operation $*$ is:

State whether the following statement is true or false. Justify: For an arbitrary binary operation $^*$ on a set $N$,$a \,^* \,a = a$ for all $a \in N$.