Galois trees in the graph of p-groups of maximal class

Alexander Cant; Heiko Dietrich; Bettina Eick; Tobias Moede

doi:10.1016/j.jalgebra.2022.03.047

Galois trees in the graph of p-groups of maximal class

Alexander Cant, Heiko Dietrich, Bettina Eick, Tobias Moede

School of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

1 Citation (Scopus)

Abstract

The investigation of the graph G_p associated with the finite p-groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green & McKay (1976–1984) introduced skeletons of G_p, described their importance for the structural investigation of G_p and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case p⩾7 and p≡5mod6, we show that they have a significant impact on the periodic patterns of G_p conjectured by Eick, Leedham-Green, Newman & O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns.

Original language	English
Pages (from-to)	429-450
Number of pages	22
Journal	Journal of Algebra
Volume	604
DOIs	https://doi.org/10.1016/j.jalgebra.2022.03.047
Publication status	Published - 15 Aug 2022

Keywords

Coclass graphs
Coclass theory
Maximal class
p-groups

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10.1016/j.jalgebra.2022.03.047Licence: CC BY

Cite this

@article{6b7157d3186e469a9659bc721ac0877d,

title = "Galois trees in the graph of p-groups of maximal class",

abstract = "The investigation of the graph Gp associated with the finite p-groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green \& McKay (1976–1984) introduced skeletons of Gp, described their importance for the structural investigation of Gp and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case p⩾7 and p≡5mod6, we show that they have a significant impact on the periodic patterns of Gp conjectured by Eick, Leedham-Green, Newman \& O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns.",

keywords = "Coclass graphs, Coclass theory, Maximal class, p-groups",

author = "Alexander Cant and Heiko Dietrich and Bettina Eick and Tobias Moede",

note = "Funding Information: This work was supported by the German Research Foundation [DFG project 386837064 to A. C. and T. M.]; and the Australian Research Council [ DP190100317 to H. D.]. Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2022",

month = aug,

day = "15",

doi = "10.1016/j.jalgebra.2022.03.047",

language = "English",

volume = "604",

pages = "429--450",

journal = "Journal of Algebra",

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T1 - Galois trees in the graph of p-groups of maximal class

AU - Cant, Alexander

AU - Dietrich, Heiko

AU - Eick, Bettina

AU - Moede, Tobias

N1 - Funding Information: This work was supported by the German Research Foundation [DFG project 386837064 to A. C. and T. M.]; and the Australian Research Council [ DP190100317 to H. D.]. Publisher Copyright: © 2022 Elsevier Inc.

PY - 2022/8/15

Y1 - 2022/8/15

N2 - The investigation of the graph Gp associated with the finite p-groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green & McKay (1976–1984) introduced skeletons of Gp, described their importance for the structural investigation of Gp and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case p⩾7 and p≡5mod6, we show that they have a significant impact on the periodic patterns of Gp conjectured by Eick, Leedham-Green, Newman & O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns.

AB - The investigation of the graph Gp associated with the finite p-groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green & McKay (1976–1984) introduced skeletons of Gp, described their importance for the structural investigation of Gp and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case p⩾7 and p≡5mod6, we show that they have a significant impact on the periodic patterns of Gp conjectured by Eick, Leedham-Green, Newman & O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns.

KW - Coclass graphs

KW - Coclass theory

KW - Maximal class

KW - p-groups

UR - https://www.scopus.com/pages/publications/85129142477

U2 - 10.1016/j.jalgebra.2022.03.047

DO - 10.1016/j.jalgebra.2022.03.047

M3 - Article

AN - SCOPUS:85129142477

SN - 0021-8693

VL - 604

SP - 429

EP - 450

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Galois trees in the graph of p-groups of maximal class

Abstract

Keywords

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Computing with Lie groups and algebras: nilpotent orbits and applications

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Galois trees in the graph of p-groups of maximal class

Abstract

Keywords

Access to Document

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Projects

Computing with Lie groups and algebras: nilpotent orbits and applications

Cite this