'Additive Inverse' in Formal Group

formal-groups

Indeed, you seem to have hit on an important property of formal groups.

If $F=F(x,y)\in R[[x,y]]$ is a (one-dimensional) formal group over the commutative ring $R$, then the set $xR[[x]]$ becomes an ordinary abelian group by means of the law of combination $F$.

That is, if $f,g\in xR[[x]]$. we may use $F$ to add the two series: $$(f+_Fg)(x)=F(f(x),g(x))\,.$$ Now it’s up to you to show that you do in fact have a group, where the inverse of $f(x)$ is $I(f(x))$.

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