If $G$ is a group of odd order made up of the units of commutative ring $R$, then we have to show that $|R|$ is of the form of $2^{k}$.
What I have figured out is if the order of the group is odd we cannot have an even order element in the group (by Lagrange theorem) so if $a\in G $ then $a\ne a^{-1} $ and $a^{-1}\in G$. Now 1 is a unit and $1\in G $ and so $-1\in G$. The order has automatically become even, we need to make it odd so that is not possible as the units will come in pair .can the group of units have an odd order?
If the above proposition is not true what counter example can I give?
The ring $R$ must have characteristic two (because $-1=1$), and hence be an algebra over the field with two elements. Being a vector space, its order must therefore be a power of two.