[Math] Subgroup of infinite group

Question

Are there any infinite order groups such that they have finite order non trivial subgroup? The simplest subgroup would be one where the element is its own inverse other than identity. Next case is if square of an element is its inverse. So are there any such infinite order groups? If such groups donot exist how to prove it?

Answer 1

The nonzero reals under multiplication have the subgroup $\{1,-1\}$.