Are there uncountably many distinct group operations on an uncountable set? Call two group operations $\ast_1$ and $\ast_2$ on a set $S$ $distinct$ if there exist $s_1,s_2\in S$ such that $s_1\ast_1 s_2 \neq s_1 \ast_2 s_2$. 
I know that there are uncountably many distinct group operations on any denumerable set. How might one show that the same is true for an uncountable set? 

Simple. You can write any infinite set $A$ as $\Bbb N\times B$ where $B$ has the same cardinality as $A$. Fix any group operation on $B$ which is "simple enough" (e.g. torsion free, or even free [abelian] group of a suitable size), and then go over all the group operations you have on $\Bbb N$ as a countable set.
Finally, $A$ is the group you get by the product of these two groups. So you just need to show that there is an uncountable collection of choices for group structure on $\Bbb N$ which give you distinct group structure on $A$. (HINT: If you chose $B$ to be torsion free this is easy enough.)
(Of course this uses the axiom of choice, but then again the statement "Every non-empty set can be made into a group" is equivalent to the axiom of choice, so it's not surprising.)
