TITLE: Groups of cardinality greater than the continuum
QUESTION [6 upvotes]: Can you give me some examples of groups of cardinality greater than continuum? All the (infinite) examples that I've been taught are countable or of cardinality continuum.
Are there any ''natural examples''? (Examples that you come across when you are researching some problem or investigating some theory)

REPLY [4 votes]: You can construct a group (ring) structure on any non empty set $E$. This is trivial if E is finite, so assume that $E$ is infinite.  The free module $Z_2[E]$ over $Z/2Z$ with basis $E$, is equipotent to $P_f(E)$ the set of all finite subsets of $E$. We have only to show that $E$ and $P_f(E)$ have the same cardinality ( so we can trnsport any structure of one of them to the other).
Now you have only to see The cardinality of the set of all finite subsets of an infinite set