Maximal linear groups induced on the Frattini quotient of a p-group

Bamberg, John; Glasby, S. P.; Morgan, Luke; Niemeyer, Alice C.

doi:10.1016/j.jpaa.2017.11.006

Cited by 10 publications

(15 citation statements)

References 21 publications

(34 reference statements)

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“…It is easy to see that ZG is not a maximal subgroup of GL(7, p), and hence Theorem 1.1 is not a consequence of the theory of Bamberg et al [2]. Note also that when p = 3, the distinct p-groups of Theorem 1.1 correspond to distinct 14-dimensional subspaces U of A 2 V. Computations in Magma [3] show, in the case of each irreducible module V, that A 2 V contains a unique 14-dimensional submodule U, with A 2 V/U V as G-modules.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

“…As we show in this paper, Z(GL(7, p))G 2 (p) is the normaliser of G 2 (p) in GL (7, p). This normaliser is not a maximal subgroup of the general linear group, and hence it does not satisfy the conditions required by Bamberg et al The methodology required to prove Theorem 1.1 is therefore different from that used in [2], and depends upon realising G 2 (p) as the automorphism group of the octonion algebra over F p . This elementary approach highlights the connection between the groups of type G 2 and this algebra, and emphasises the uniqueness of these groups among the other groups of Lie type.…”

Section: Introductionmentioning

confidence: 99%

“…As in the traditional case, it is natural to consider simple groups, since this may lead to an understanding of the problem for finite groups in general. In this paper, we begin a programme to study the nonabelian analogue for the exceptional groups of Lie type; the classical groups of Lie type were studied in [2]. The groups of type G 2 are the nontwisted exceptional groups of lowest Lie rank, and so we focus first on those.…”

Section: Introductionmentioning

confidence: 99%

“…Bryant and Kovács [5] proved that if H is any subgroup of GL(d, p), with d > 1, then there exists a p-group P of rank d such that A(P) = H. From the proof of their result, we see that both the class and exponent of the group P are roughly |GL(d, p)| (depending on both d and p), and the order of P is not explicit, but must be enormous. Bamberg et al [2] were inspired to investigate the necessity of the large exponent and class of the groups constructed in [5]. They showed that if H satisfies a few extra conditions, then there exists a p-group P with A(P) = H, such that P has rank d; nilpotency class two, three or four; exponent p; and order at most p d 4 /2 .…”

Section: Introductionmentioning

confidence: 99%

“…Our method of proof for Theorem 1.1 is to appeal to a description of groups of class two and exponent p, as in [6,, [8,Ch. 9.4] and [15,Section 2] (see also [2,Section 2]). Such a p-group P is a quotient of the universal p-group of nilpotency class two, rank d and exponent p, that is, the largest finite group with these properties.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On -Groups With Automorphism Groups Related to the Chevalley Group

Bamberg¹,

Freedman²,

Morgan³

2020

J. Aust. Math. Soc.

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Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\text{Aut}(P)$ induces $N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$ on the Frattini quotient $P/\unicode[STIX]{x1D6F7}(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p=3$ our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_{2}(q)$ on the octonion algebra over $\mathbb{F}_{q}$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_{q}[G_{2}(q)]$-module.

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Section: Proof Of the Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On -Groups With Automorphism Groups Related to the Chevalley Group

Bamberg¹,

Freedman²,

Morgan³

2020

J. Aust. Math. Soc.

Self Cite

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Standortübergreifende Forschung zu Carbonbetonstrukturen im SFB/TRR 280

et al. 2021

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Neue Materialien ermöglichen neue Bauformen und Konstruktionsarten. Erste Bauprojekte mit den neuen Materialien zeigten aber, dass zunächst weiterhin nach traditionellen – im Falle von Carbonbeton dem Stahlbeton entlehnten – Konstruktionsprinzipien gebaut wird, herkömmliche Materialien also lediglich substituiert werden. Erst in Verbindung mit intelligenten Konstruktionsstrategien wird das volle Potenzial innovativer Werkstoffe ausgenutzt. Losgelöst von etablierten Denkmustern sollen im Sonderforschungsbereich/Transregio (SFB/TRR) 280 „Konstruktionsstrategien für materialminimierte Carbonbetonstrukturen – Grundlagen für eine neue Art zu bauen“ an der Technischen Universität Dresden und der RWTH Aachen University die Grundlagen für eine neue Form des Bauens mit Beton auf Basis tiefgreifender Erkenntnisse zum strukturmechanischen Verhalten neuartiger, mineralisch basierter Komposite geschaffen werden. Die neuen Leichtbau‐Konstruktionsstrategien und Werkstoffkombinationen reduzieren Ressourcen‐ und Energieverbrauch bei gleichzeitig hoher Gebrauchstauglichkeit, Tragsicherheit und Dauerhaftigkeit; eine anspruchsvolle Ästhetik leistet einen Beitrag zur Baukultur. Der langjährige Forschungsverbund von TU Dresden und RWTH Aachen vereint exzellente Kompetenzen, was die Erforschung des materialminimierten Bauens mit mineralischen Kompositen beflügeln wird. Im Beitrag werden die für den Zeitraum Juli 2020 bis Juni 2024 geplanten Forschungsaktivitäten vorgestellt.

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Collaborative research on carbon reinforced concrete structures in the CRC/TRR 280 project

Beckmann

Bielak

Bosbach

et al. 2021

Civil Engineering Design

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New materials allow new building designs and construction types. However, initial construction projects using the new materials show that traditional construction principles ‐ in the case of carbon reinforced concrete those derived from steel reinforced concrete ‐ are still being used. In other words, conventional materials are merely substituted. Only combined with intelligent construction strategies it is possible to exploit the full potential of innovative materials. Detached from established patterns of thought, the fundamentals for a new way of building with concrete are to be created in the frame of the Collaborative Research Centre Transregio (CRC/TRR) 280 project “Design Strategies for Material‐Minimised Carbon Reinforced Concrete Structures–Principles of a New Approach to Construction” at Technische Universität Dresden (TUD) and RWTH Aachen University. These are to be based on profound insights into the mechanical behavior of novel mineral structures. Innovative lightweight construction strategies and material composites reduce resource and energy consumption while maintaining high levels of usability, structural safety, and durability, while the aspirational esthetics make a valuable contribution to the culture of building. The long‐term research alliance of TUD and RWTH pools the excellent competencies, inspiring research into material‐minimized construction with mineral composites. The article highlights the research activities planned for the first period from July 2020 to June 2024.

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Maximal linear groups induced on the Frattini quotient of a p-group

Cited by 10 publications

References 21 publications

On -Groups With Automorphism Groups Related to the Chevalley Group

On -Groups With Automorphism Groups Related to the Chevalley Group

Standortübergreifende Forschung zu Carbonbetonstrukturen im SFB/TRR 280

Collaborative research on carbon reinforced concrete structures in the CRC/TRR 280 project

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