There are groups of any size : (except $0$): for finite ones, you can always take the cyclic groups. For infinite ones, to get a group of cardinality $\kappa\geq\aleph_0$, just take the direct sum of $\kappa$ copies of $\mathbb{Z}$. Given a set $S$ of cardinality $\lambda$ (finite or infinite), pick a bijection $f$ from $S$ to a group $G$ of cardinality $\lambda$, and define the operation on $S$ by $a*b = f^{-1}(f(a)f(b))$.
There are groups of any size: for finite ones, you can always take the cyclic groups. For infinite ones, to get a group of cardinality $\kappa\geq\aleph_0$, just take the direct sum of $\kappa$ copies of $\mathbb{Z}$. Given a set $S$ of cardinality $\lambda$ (finite or infinite), pick a bijection $f$ from $S$ to a group $G$ of cardinality $\lambda$, and define the operation on $S$ by $a*b = f^{-1}(f(a)f(b))$.