Continuous automorphisms and an equivalence relation In \(K[[X]]\)

Let K be an arbitrary commutative field and let \(R = K[[X]]\) be the ring of formal power series in one variable. Let \(G_{R}\) be the set of all power series of the form \(u = Xv\), where \(v\) is a unity in \(R\). Relative to the usual composition \(G_{R}\) becomes a topological group with respect to the \(X-\)adic topology of \(R\). We also study an equivalence relation on \(R\). Let \(R = K[[X]]\) be the ring of formal power series in one variable over a fixed commutative field \(K\). We denote by \(ord f = min \{i: a_{i} ≠ 0\} \) for any \(  f ∈ R\) . It is well known that \(ord f\) is a valuation on \(R\) and \(R\) becomes a complete topological ring relative to the topology induced by this valuation. Let  \(G_{G}=\{u∈ R: ord u=1\}\) and for \(u,v ∈ R_{G}\) we denote \((uov)(X)=v(u(X))\), a new element of \(R_{G}\).