= Prime-to-residue-characteristic multiplication on a formal group
{title2=$[n]_F:F(\mathfrak m)\xrightarrow{\sim}F(\mathfrak m),\quad p\nmid n$}
Let $R$ be a complete <discrete valuation ring> of residue <characteristic of a field> $p$, with <maximal ideal> $\mathfrak m$, and let $F$ be a one-dimensional <formal group law> over $R$. Its multiplication series is $[n]_F(T)=nT+O(T^2)$. If $p\nmid n$, the <invertible morphism criterion for formal group laws> constructs its inverse with coefficients in $R$. Both <formal power series> converge on $\mathfrak m$, since their coefficients are integral and the powers of an element of $\mathfrak m$ tend to zero. Their compositions are the identity there. Thus multiplication by $n$ is bijective, including when $p=2$ and $n$ is odd. Every finite-order element of $F(\mathfrak m)$ consequently has <order of a group element> equal to a <prime power> with prime $p$.
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