Explicit embeddings of finite abelian \(p\)-groups in the group \(\mathcal J(\mathbb F_p)\).

*(English. Russian original)*Zbl 1329.20036
Math. Notes 97, No. 1, 63-68 (2015); translation from Mat. Zametki 97, No. 1, 74-79 (2015).

The author investigates finite abelian subgroups of the Jennings group \(\mathcal J(\mathbb F_p)\), also commonly known as the Nottingham group and denoted \(\mathcal N\). Elements of \(\mathcal J(\mathbb F_p)\) are normalised formal power series in \(\mathbb F_p[[x]]\), that is formal power series of the form
\[
f(x)=x+a_1x^2+a_2x^3+\cdots
\]
with the group operation being substitution. Alternatively the elements can be viewed as automorphisms of the field of finitely tailed Laurent series over \(\mathbb F_p\), denoted \(\mathbb F_p((x))\).

The Jennings group is a finitely presented pro-\(p\) group in which every countably based pro-\(p\) group can be embedded. However, finding explicit embeddings of finite \(p\)-groups has proved difficult. The case of elements of order \(p\) is understood [B. Klopsch, J. Algebra 223, No. 1, 37-56 (2000; Zbl 0965.20021)] and in this case closed formulae are given. Recently J. Lubin has investigated embeddings of finite cyclic groups [Bull. Lond. Math. Soc. 43, No. 3, 547-560 (2011; Zbl 1267.11115)]. In this paper the author gives explicit embeddings of finite abelian groups where explicit is interpreted to mean that the generators of the group can be evaluated to arbitrarily high degree in finitely many steps.

The author begins by constructing totally ramified extensions \(k_n\) of the field \(k_0=\mathbb F_p((t))\) such that every finite abelian totally ramified extension of \(k_0\) is contained in \(k_n/k_0\) for some \(n\), such an extension is called a Lubin-Tate extension. Then, to embed a finite abelian group \(A\) in the Jennings group first the group \(C\cong C_{p^{m-1}}\times\cdots\times C_{p^{m-m/2}}\) is embedded in the Galois group of \(k_n/k_0\) where \(n=p^m\) and \(m\) is even and sufficiently large. Taking appropriate powers of the generators of \(C\), which are explicitly known, gives an embedding of \(A\) in the Galois group. Finally, noting that \(k_n\) is a totally ramified extension of \(k_0\) gives that \(k_n\cong\mathbb F_p((t_n))\) for a uniformizer \(t_n\), and thus automorphisms of \(k_n\) are elements of the automorphism group of \(\mathbb F_p((t_n))\) of the required form.

The Jennings group is a finitely presented pro-\(p\) group in which every countably based pro-\(p\) group can be embedded. However, finding explicit embeddings of finite \(p\)-groups has proved difficult. The case of elements of order \(p\) is understood [B. Klopsch, J. Algebra 223, No. 1, 37-56 (2000; Zbl 0965.20021)] and in this case closed formulae are given. Recently J. Lubin has investigated embeddings of finite cyclic groups [Bull. Lond. Math. Soc. 43, No. 3, 547-560 (2011; Zbl 1267.11115)]. In this paper the author gives explicit embeddings of finite abelian groups where explicit is interpreted to mean that the generators of the group can be evaluated to arbitrarily high degree in finitely many steps.

The author begins by constructing totally ramified extensions \(k_n\) of the field \(k_0=\mathbb F_p((t))\) such that every finite abelian totally ramified extension of \(k_0\) is contained in \(k_n/k_0\) for some \(n\), such an extension is called a Lubin-Tate extension. Then, to embed a finite abelian group \(A\) in the Jennings group first the group \(C\cong C_{p^{m-1}}\times\cdots\times C_{p^{m-m/2}}\) is embedded in the Galois group of \(k_n/k_0\) where \(n=p^m\) and \(m\) is even and sufficiently large. Taking appropriate powers of the generators of \(C\), which are explicitly known, gives an embedding of \(A\) in the Galois group. Finally, noting that \(k_n\) is a totally ramified extension of \(k_0\) gives that \(k_n\cong\mathbb F_p((t_n))\) for a uniformizer \(t_n\), and thus automorphisms of \(k_n\) are elements of the automorphism group of \(\mathbb F_p((t_n))\) of the required form.

Reviewer: Rachel D. Camina (Cambridge)

##### MSC:

20E18 | Limits, profinite groups |

20D15 | Finite nilpotent groups, \(p\)-groups |

20E07 | Subgroup theorems; subgroup growth |

11S15 | Ramification and extension theory |

##### Keywords:

finite Abelian groups; Jennings group; Nottingham group; commutative rings; formal power series; finitely presented pro-\(p\) groups; explicit embeddings; totally ramified field extensions
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\textit{D. D. Kiselev}, Math. Notes 97, No. 1, 63--68 (2015; Zbl 1329.20036); translation from Mat. Zametki 97, No. 1, 74--79 (2015)

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##### References:

[1] | Jennings, S A, Substitution groups of formal power series, Canad. J. Math., 6, 325-340, (1954) · Zbl 0058.02201 |

[2] | Camina, R, Subgroups of the Nottingham group, J. Algebra, 196, 101-113, (1997) · Zbl 0883.20015 |

[3] | Babenko, I K, Algebra, geometry, and topology of the substitution group of formal power series, Uspekhi Mat. Nauk, 68, 3-76, (2013) |

[4] | Lubin, J; Tate, J, Formal complexmultiplication in local fields, Ann. ofMath. (2), 81, 380-387, (1965) · Zbl 0128.26501 |

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