# The Torsion Subgroup of an Abelian Group

Posted by Jason Polak on 19. December 2015 · Write a comment · Categories: group-theory, homological-algebra · Tags:

Let $A$ be an abelian group. We call an element $a\in A$ torsion if there exists a natural number $n$ such that $na = 0$. The set of all torsion elements $T(A)$ of $A$ form a subgroup of $A$, and we can think of $T$ as an endofunctor on the category of abelian groups. Here are some examples:

• $T(\Z) = 0$
• $T(\Z\oplus \Z/n) = 0\oplus\Z/n\cong\Z/n$
• $T(\Q) = 0$
• $T(\Q/\Z) = \Q/\Z$

Finding the set of torsion points of an abelian group isn’t always easy as in these examples, since abelian groups may not always be written out in such an explicit way. A fascinating and nontrivial result of Barry Mazur is that the torsion subgroup of $E(\Q)$ for an elliptic curve $E$ over $\Q$ is one of fifteen possibilities: $\Z/n$ for $1\leq n\leq 10$ or $n=12$ or $\Z/2\times \Z/2n$ for $1\leq n\leq 4$. Such strange numerology!
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