Is $\{0,1\}$ a subgroup of $\mathbb{R}$ under multiplication? I am playing around with subgroups and I was wondering if the group $\{0,1\}$ is a subgroup of $\mathbb{R}$ with respect to multiplication. It seems to  fit the criteria but the inverse property is making me worry because $0$ is its own inverse. I vaguely remember a proof that if an element was its own inverse than it must be the identity element  but $1$ would need to be the identity here. Thanks in advance! 

Hint: The Cayley table of $\{0, 1\}$ under multiplication is
$$\begin{array}{c| c c}
\times & 0 & 1\\
\hline
0 & 0 & 0 \\
1 & 0 & 1.
\end{array}$$
What do you know about Latin squares?
