I've always wondered, and I can't seem to find out...
well i presume you can't find out because you are assuming it is an algebra term, and nowadays algebra is absurdly isolated from its roots, often in topology or geometry.
ironically i was just reading miles reid's essay in undergraduate commutative algebra, on how algebra for some reason has become highly abstract and separated from the rest of mathematics, making it more difficult to teach, and for some reason also harder to make jokes in the subject.
I am having trouble finding a good reference for you online, but you might look at the notes "homology" by jesper moller on his homepage, (not the statistician) headed "from singular homology to alexander duality" or some such.
look in the section on cellular homology of cell complexes, page 30 of the current version on his webpage today, for the formulas for homology of certain spaces formed by attaching cells.
now i admit this is just a guess, but it is an informed one. It seems the term torsion arose in topology, to describe the invariants of homology groups of cell complexes, especially surfaces, in the early part of the past century.
i.e. the way to analyze the topology of a surface is to realize that a surface is formed by attaching cells via maps of their boundaries, to a simpler space one already has.
E.g. to form a sphere, one attaches a 2 disk to a point by a constant map that collapses the boundary of the 2 disc to that point.
here there is no torsion, which everyone knows means twisting.
to form a projective plane on the other hand one starts from a circle and attaches a 2 disc by a map that wraps the boundary of the disc twice around the circle.
this torsion or twisting of the boundary produces the term Z/2 in the 1st homology group, i.e. H1 of the new space is H1 of the circle modulo the image of the map on H1 of the boundary of the disc, induced by the attaching map. this is the formula on page 30 of mollers notes.
i.e. attaching a cell kills homology in one dimension down, but only kills those cycles in the image of the attaching map.
more generally mapping the boundary circle in by the map z goes to
z^n, produces a summand of Z/n in the 1st homology.
this is the obvious meaning of torsion in homology groups, presumably the historical one.
they used to be called torsion coefficients, before homology of spaces was formalized as a group.
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