In 2008 Dilworth, Odell, Schlumprecht, and Zs\'ak proved that every separable Banach space contains a net that is an (additive) subgroup. In this talk we will explain how a modification of an argument due to Victor Klee permits to obtain a shorter self-contained proof of the said result and also to extend the result to non-separable Banach spaces. In the second part of the talk, we focus on the non-separable Hilbert space $\ell_2(\Gamma)$, for $|\Gamma|=\mathfrak{c}$, and show that it contains a subgroup which is $(\sqrt{2}+)$-separated and $1$-dense. We also explain how this construction answers a problem originating from the theory of tilings. Joint work with Carlo Alberto De Bernardi and Jacopo Somaglia.