We know that any finite group can't be isomorphic to any of its proper subgroups.
Some countably infinite groups, like $\mathbb{Z}$, do have this property of course, as $\mathbb{Z} \cong 2\mathbb{Z}$ . Could we do something like this for an $\mathbb{R}$? This raises some questions for me:
$1.$ Is there an obvious example of an infinite group that is not isomorphic to any of its proper subgroups?
$2.$ Is there an easy criterion to establish whether an infinite group does or does not have this property?
An obvious example is the additive group $\mathbb Q$. You may be interested in this http://www.maa.org/sites/default/files/269079615024.pdf
This article from Mathematics Magazine (vol. 72, no. 5, December 1999, p. 388) is "On Groups That Are Isomorphic to a Proper Subgroup" by Shaun Fallat, Chi-Kwong Li, David Lutzer, and David Stanford (College of William and Mary, Williamsburg VA 23187-8795).