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A power-commutator presentation for the group of truncated polynomials under substitution
Abstract
Given a prime p, the Nottingham group G(p) is the group of formal power series x+a2x2+a3x3+⋯, ai∈bf Z/pbf Z, under substitution. For n≥ 1, we write Gn(p) for the quotient of G(p) by its normal subgroup Kn={x+an+1xn+1+an+2xn+2+⋯ | ai∈ bf Z/pbf Z}, namely the group of truncated polynomials x+a2x2+⋯+anxn, ai∈ bf Z/pbf Z, under substitution, which is a p-group of order pn−1 generated by x+x2,…,x+xn. In this paper, we determine the power-commutator presentation of G15(p) relative to x+x2,…,x+x15. This, in turn, automatically gives the power-commutator presentation of Gn(p) relative to x+x2,…,x+xn for any 1≤ n<15, as well as the initial segments of the power and commutator relations in Gn(p) corresponding to x+x2,…,x+x15 for any n>15.
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BibTeX
@misc{szechtman2026powercommutator,
title = {{A power-commutator presentation for the group of truncated polynomials under substitution}},
author = {Fernando Szechtman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01469},
eprint = {2610.01469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01469},
}
RIS
TY - UNPB
TI - A power-commutator presentation for the group of truncated polynomials under substitution
AU - Fernando Szechtman
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01469
UR - https://arxiv.org/abs/2610.01469
AB - Given a prime p, the Nottingham group G(p) is the group of formal power series x+a_2x^2+a_3x^3+⋯, a_i∈bf Z/pbf Z, under substitution. For n≥ 1, we write G_n(p) for the quotient of G(p) by its normal subgroup K_n={x+a_n+1x^n+1+a_n+2x^n+2+⋯ | a_i∈ bf Z/pbf Z}, namely the group of truncated polynomials x+a_2x^2+⋯+a_nx^n, a_i∈ bf Z/pbf Z, under substitution, which is a p-group of order p^n−1 generated by x+x^2,…,x+x^n. In this paper, we determine the power-commutator presentation of G_15(p) relative to x+x^2,…,x+x^15. This, in turn, automatically gives the power-commutator presentation of G_n(p) relative to x+x^2,…,x+x^n for any 1≤ n<15, as well as the initial segments of the power and commutator relations in G_n(p) corresponding to x+x^2,…,x+x^15 for any n>15.
ER -