= Depth filtration of the Nottingham group
{title2=$\mathcal N_m=\{f:f(t)\equiv t\pmod{t^{m+1}}\}$}
Put $\mathcal N_m=\{f:f(t)\equiv t\pmod{t^{m+1}}\}$ for $m\ge1$. These are <open normal subgroups> and $\mathcal N_m/\mathcal N_{m+1}\cong(\mathbb F_p,+)$ via the coefficient of $t^{m+1}$. Thus $[\mathcal N:\mathcal N_m]=p^{m-1}$. The depth of a nonidentity series is $\operatorname{ord}_t(f(t)-t)-1$. If $f=t+at^{r+1}+\cdots$ and $g=t+bt^{s+1}+\cdots$, expansion of substitution gives
$$
[f,g](t)=t+ab(r-s)t^{r+s+1}+O(t^{r+s+2}).
$$
Here $[f,g]=f^{-1}\circ g^{-1}\circ f\circ g$. This formula controls <commutator subgroups> and <normal subgroups>.
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