TITLE: Order of a group with all elements as self-inverse
QUESTION [5 upvotes]: It is a common result that, if all elements of a group are self-inverse, then the group is Abelian. Is it necessarily finite also?

REPLY [2 votes]: It doesn't have to be finite. Indeed, for any cardinal $\kappa$, there's a group with that property of order $2^\kappa$:
Be $M$ any set of cardinality $\kappa$. Then $(\mathcal P(M),\triangle)$ is a group with this property, where $\mathcal P(M)$ is the power set of $M$, and $\triangle$ denotes the symmetric set difference.
Indeed, if you allow groups to be defined not only on sets, but also on proper classes, the nimbers with nimber addition form a group with this property whose order is the cardinality of the ordinals.