Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
Torsion is used to refer to elements of finite order under some binary operation. It doesn't seem to bear any relation to the ordinary everyday use of the word or with its use in differential geometry (which relates back to the ordinary use of the word). So how did it acquire this usage in algebra?
I'm interested to understand the intuition behind why the word "torsion" was chosen for this notion, as well as when it was first used.
John Stillwell wrote that "the word 'torsion'entered the theory of abelian groups as a result of the derivation of the one-dimensional torsion coefficients by abelianization of the fundamental group in Tietze 1908" [Classical Topology and Combinatorial Group Theory, 1993, Sec. 5.1.1, p. 170]. Below is an excerpt providing further context.
The appropriate notions of "sum" and "boundary,"and the correct choice of k-dimensional manifolds admissible as basis elements, were found only after considerable trial and error. "Appropriate" initially meant satisfying the relation $B_k = B_{m-k}$ since this was the relation Poincare tried to prove in his 1895 paper. Heegaard 1898 showed this work to be in error by constructing a counterexample. Poincare then changed the definition and proved the theorem again in Poincare 1899, inventing the tool of simplicial decomposition for the purpose. He also made a thorough analysis of his error, uncovering the important concept of torsion in Poincare 1900, and exposing the breakdown of his earlier proof as failure to observe torsion.
Torsion is present when an element a does not form a boundary taken once, but does when taken more than once. An example is the curve $a$ in the projective plane $P$ which generates $\pi_1(P).$ Then $a^2$ is the boundary of a disc, though a itself does not separate $P.$ Poincare justified the term "torsion" by showing that $(m-1)$-dimensional torsion is present only in an $m$-manifold which is nonorientable, and hence twisted onto itself in some sense.
In his first topology paper, Poincare 1892 showed that the Betti numbers alone did not determine a manifold up to homeomorphism. By 1900 he was hoping that torsion numbers would supply the missing information, and his paper of that year contains a decomposition of the homology in- formation in each dimension $k$ into the Betti number $B_k$ and a finite set of numbers called $k$-dimensional torsion coefficients. Since Noether 1926 it has been customary to encode this information in an abelian group $H_k$ called the $k$-dimensional homology group, and Poincare's construction can in fact be seen as the decomposition of a finitely generated abelian group into cyclic factors (see the structure theorem 5.2). The word "torsion," which appears so inexplicably in most algebra texts, entered the theory of abelian groups as a result of the derivation of the one-dimensional torsion coefficients by abelianization of the fundamental group in Tietze 1908 (see 5.1.3. and 5.3).
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!
From Rotmans book "Advanced Modern Algebra":
This terminology comes from algebraic topology. To each space $X$, a sequence of abelian groups is assigned, called homology groups, and if $X$ is “twisted,” then there are elements of finite order in some of these groups.
