Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S\_\omega$, the group of permutations of the set of non-negative integers $\omega$?



Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S_\omega$?
See Theorem 4.3 of [this paper](https://pure.tue.nl/ws/portalfiles/portal/1866721/597514.pdf) by De Bruijn. Any abelian group of order $2^\kappa$ can be embedded in $Sym(\kappa)$ when $\kappa$ is infinite. (There is also an [addendum](https://core.ac.uk/download/pdf/82202896.pdf) to the paper which corrects some error in the proof.)


