# Where does the word “torsion” in algebra come from?

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).