= Torsion subgroup killed by an integer
{title2=$G[m]=\ker(m:G\to G)$}
= m-torsion subgroup
{synonym}
= Torsion kernel of multiplication
{synonym}
For an <abelian group> $G$ and an integer $m\geq1$, its $m$-torsion subgroup is $G[m]=\{x\in G:mx=0\}$, the kernel of multiplication by $m$. In a <finite abelian group>, every nonempty fiber of this homomorphism has <cardinality> $|G[m]|$, so $|G:mG|=|G[m]|$. This follows directly: if $mx=my$, then $x-y\in G[m]$, and conversely translating by an element of $G[m]$ leaves the image unchanged. On the <cyclic group> $\mathbb Z_N$, the solutions of $mx=0$ number $\gcd(m,N)$; divide both integers by their greatest common divisor to see that $x$ must be a multiple of $N/\gcd(m,N)$.
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