Powers of elements of the series substitution group J(Z2)J(Z2)

For the Jennings group J(k)J(k) of substitutions of formal power series with coefficients in a field k of positive characteristic (the Nottingham group), the depth of the powers of its elements is studied. In particular, it is shown that the case of a field with characteristic 2 is completely different from the case of a field with odd prime characteristic. It is also shown that the case of the field k=Z2k=Z2 differs from the case of other fields with characteristic 2. Explicit embeddings of the groups ZpmZpm and Zp⊕ZpZp⊕Zp in J(Zp)J(Zp) are constructed.