Groups with elements with large finitie order are equal to groups How to prove that every group possesing elements of arbitarily large finitie order is equivalent to a group in which there is an element of infinitie oreder?
As equivalent i mean elementary equivalent structures
Two structures M and N of the same signature σ are elementarily equivalent if every first-order sentence (formula without free variables) over σ is true in M if and only if it is true in N

A clean way to do this is to choose (I hope, you are OK with the AC, otherwise, I cannot help you) a nonprincipal ultrafilter $\omega$ on ${\mathbb N}$ and, given your group $G$ (with elements of arbitrarily high finite order), consider the ultraproduct
$$
H:= \left( \prod_{n\in {\mathbb N}} G\right)/\omega.
$$
By Los' Theorem, $G$ and $H$ are elementary-equivalent. A the same time, $H$ will have an element of infinite order. I will let you figure out which one. 
