• 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).
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