There is an entry here at the "Earliest known uses" site that is sometimes useful. Quoting directly from it, for readers' benefit:
TORSION as used in group theory: an element of a group G is a torsion element if it generates a finite subgroup of G. An abelian group consisting entirely of torsion elements is called a torsion group. In any abelian group, the torsion elements form a subgroup, frequently called the torsion subgroup of G. An abelian group is torsion-free if the neutral element is its only torsion element.
This terminology seems to have arisen around 1930. Its origin lies in algebraic (or combinatorial) topology. Poincaré (Second complément à l´Analysis Situs, Proc. London Math. Soc, vol. 32 (1900), 277-308) defined torsion coefficients for manifolds (variétés), and he distinguished manifolds with and without torsion. In a later terminology, his torsion coefficients are structure constants of homology groups. In 1935, the textbook Topologie I by Alexandroff-Hopf has the following concept of torsion: “The elements of finite order of the r-th Betti group of E form a subgroup called the r-th torsion group of E.” Here, the use of the word torsion group is still tied to a topological context even though the concept of a torsion subgroup seems to be hinted at. In the group theoretical chapter of this book, the word torsion is not used. But in the same year 1935, the paper Countable torsion groups by Leon Zippin (Annals of Math. 36, 86-99) contains the following definition: “A torsion group T is a discrete countable abelian group, every element of which is of finite order” - quite the modern definition, except for the restriction to countable groups. Two years before, Ulm (Zur Theorie der abzählbar-unendlichen Abelschen Gruppen, Math. Annalen 107, pp. 774-803) did not yet use these terms, writing instead “groups all of whose elements have finite exponents.” The torsion terminology was slow in obtaining general acceptance; it was frequently used in research papers in the 1940s, and is applied consistently in Kaplansky’s Infinite Abelian Groups of 1954, while on the other hand, Marshall Hall’s textbook on group theory, published in 1959, introduces the term “periodic group,” explaining that “the term torsion group is used in certain applications.” The influential book on topological groups, Abstract Harmonic Analysis I by Hewitt and Ross (1963) uses the torsion terminology and may have been important in promoting it.
[This entry was contributed by Peter Flor.]
End quote. It's an interesting site!