| Autor: |
Cant, Alexander, Dietrich, Heiko, Eick, Bettina, Moede, Tobias |
| Rok vydání: |
2021 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
The investigation of the graph $\mathcal{G}_p$ 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 and McKay (1976-1984) introduced skeletons of $\mathcal{G}_p$, described their importance for the structural investigation of $\mathcal{G}_p$ 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 $p \geq 7$ and $p \equiv 5 \bmod 6$, we show that they have a significant impact on the periodic patterns of $\mathcal{G}_p$ conjectured by Eick, Leedham-Green, Newman and O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns. |
| Databáze: |
arXiv |
| Externí odkaz: |
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