When is a group isomorphic to a proper subgroup of itself?

Question

A infinitely generated additive group G and its subgroup K, when they are isomorphic to each other? Is there any theorem on this?

Answer 1

Take the set of polynomials in a variable $x$ with integer coefficients, it is an infinitely generated group on the generating set $\{1,x,x^2,x^3,\ldots\}$. Take the proper subgroup of polynomials involving only the even powers: It is a proper subgroup isomorphic to the whole group (it is infact isomorphic even as a ring).

Answer 2

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.