Let $X$ be an infinite set. What I'm looking for is a lower bound (or even better, precise cardinal number) of possible non-isomorphic group structures on $X$. Since every group is built on some function $X\times X\to X$ then there is at most $|X|^{|X\times X|}$ group structures on $X$. If I remember correctly this number is equal to $2^{|X|}$ for infinite $X$.
What about a lower bound? What I have so far is: if $X$ is infinite then the free group $\mathbb{F}(X)$ generated on $X$ is of order $|X|$. Furthermore $G\times\mathbb{F}(X)$ is still of order $|X|$ if $|G|\leq |X|$. And if $G, H$ are finite then $G\times\mathbb{F}(X)\simeq H\times\mathbb{F}(X)$ if and only if $G\simeq H$. In particular there is at least countably many group structures on $X$. But this lower bound is quite bad. I suspect that there's at least $|X|$ group structures on $X$. And that would be a good enough lower bound for me.
If the general case is too difficult then putting $X:=\mathbb{R}$ will be good as well.
Can someone help me with the proof? Or give ma a hint at least?