• EVERY FINITELY PRESENTED TORSION GROUP IS FINITE

Abstract

Four lemmas are requisite to the presented proof that every finitely presented torsion group is finite (equivalently, some non-finitely presented torsion group  is infinite), namely,  is an abelian subgroup of the real numbers modulo 1 (proved using the Subgroup Criterion), every element of  is of finite order (equivalently, some element not in  is of infinite order),  is not finitely presented, and  is infinite; all of which are shown in this paper. The proof that some element not in  is of infinite order relies on the proof that the sum of a rational number and an irrational number is equal to an irrational number (Khan, n.d.). In the proof that  is not finitely presented, we first show that  is not finitely generated utilizing the important proposition that states that “the subgroup generated by the set  is equal to closure of ” (Dummit & Foote, 2004, p. 63).