TITLE: When is a group isomorphic to a proper subgroup of itself?
QUESTION [7 upvotes]: A infinitely generated additive group G and its subgroup K, when they are isomorphic to each other? Is there any theorem on this?

REPLY [7 votes]: There is no any general theorem to solve your problem
Clearly if a group is isomorphic to a proper subgroup of itself then $|G|$ must be infinite, but being infinite is not sufficient to be isomorphic to a proper subgroup of itself, there exist some groups which are infinite and isomorphic to a subgroup of themselves such as $\Bbb Z$ under addition (isomorphic to the group of even integers) and $\Bbb R^+,\Bbb R^*,\Bbb Q^+,\Bbb Q^*$ under multiplication. and there are some infinite groups which are not isomorphic to a subgroups of themselves such that $\Bbb Q,\Bbb Q/\Bbb Z$ under addition.