Please help.....
For the following sets, with the binary operation given, determine whether or not it forms a group, by checking the group axioms.
(R,◦),where x◦y=(xy)/√3
What are the group axioms? First, you need an identity element: $\displaystyle e\circ x = x\circ e = \dfrac{ex}{\sqrt{3}} = x$. By cancellation, we find $\displaystyle e = \sqrt{3}$. Next, check that associativity holds:
$\displaystyle \begin{align*}x\circ (y\circ z) & = \dfrac{ x \tfrac{yz}{ \sqrt{3} } }{ \sqrt{3} } \\ & = \dfrac{ \tfrac{xy}{\sqrt{3}} z }{ \sqrt{3} } \\ & = (x\circ y)\circ z\end{align*}$
This follows from associativity and commutativity of real numbers.
Next, check that each element has an inverse. Let $\displaystyle x \in \mathbb{R}$. You want to find $\displaystyle y \in \mathbb{R}$ so that $\displaystyle x\circ y = y\circ x = e = \sqrt{3}$. So, $\displaystyle x\circ y = \dfrac{xy}{\sqrt{3}} = \sqrt{3}$. Solving for $\displaystyle y,$ we find $\displaystyle y = \dfrac{3}{x}$. Hence, $\displaystyle y$ exists so long as $\displaystyle x \neq 0$.
So, no, it is not a group, but $\displaystyle \left(\mathbb{R}\setminus \{0\},\circ \right)$ is a group.