Completing Segre's proof of Wedderburn's little theorem

Bamberg, John; Penttila, Tim

doi:10.1112/blms/bdv021

Cited by 6 publications

(7 citation statements)

References 21 publications

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“…The results contained in this paper are in a similar vein to the results by Bamberg and Penttila [2] and Penttila and Siciliano [22] (proving that a finite Bol field of even order is a nearfield without using the Feit-Thomson theorem that group of odd order are soluble), both also obtained using the underlying connections between the three areas.…”

Section: Introductionsupporting

confidence: 82%

“…Since u + u ∈ GL(V ), the kernel of u and the kernel of u meet just in {0}, so, by dimensions, the image of u equals to kernel of u . But now (2) shows that the image of u is γ−invariant for all γ ∈ α ∪ β . Thus the image of u is α, β −invariant.…”

Section: Background On Alternative Division Ringsmentioning

confidence: 87%

“…In 1905, Wedderburn [29] proved that a finite (associative) division ring is a field -a result now known as Wedderburn's little theorem. A recent proof was given by Bamberg and Penttila [2], which exploited the connection with geometry -that the theorem is equivalent to the statement that a finite Desarguesian projective space is Pappian. Artin proved a stronger theorem, first published by his student Zorn in 1931 [30] and now known as the Artin-Zorn theorem: a finite alternative division ring is a field.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Variations on a Theme of Glauberman

Penttila

Siciliano

2020

Results Math

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Section: Introductionsupporting

confidence: 82%

Section: Background On Alternative Division Ringsmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

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Variations on a Theme of Glauberman

Penttila

Siciliano

2020

Results Math

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“…Secondly, we are interested in a partial automation of many steps in the Desargues' theorem presented in [23,24] using mainly the submodularity. To extend this analysis, we will examine two other consequent theorems in projective incidence geometry: Dandelin-Galluci theorem [1,2] and the harmonic conjugate. The aim will be to make proof as readable as possible by removing technical details, thus being as close as possible to a mathematical proof.…”

Section: Discussionmentioning

confidence: 99%

Two cryptomorphic formalizations of projective incidence geometry

Braun

Magaud

Schreck

2018

Ann Math Artif Intell

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Incidence geometry is a well-established theory which captures the very basic properties of all geometries in terms of points belonging to lines, planes, etc. Moreover, projective incidence geometry leads to a simple framework where many properties can be studied. In this article, we consider two very different but complementary mathematical approaches formalizing this theory within the Coq proof assistant. The first one consists of the usual and synthetic geometric axiom system often encountered in the literature. The second one is more original and relies on combinatorial aspects through the notion of rank which is based on the matroid structure of incidence geometry. This paper mainly contributes to the field by proving the equivalence between these two approaches in both 2D and 3D. This result allows us to study the further automation of many proofs of projective geometry theorems. We give an overview of techniques that will be heavily used in the equivalence proof and are generic enough to be reused later in yet-to-bewritten proofs. Finally, we discuss the possibilities of future automation that can be envisaged using the rank notion.

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“…The proposed method would be more algebraically flexible if it had two operations (+ and •) with the distributive property over the finite set of elements. Unfortunately, it is theoretically impossible to define a non-commutative finite division ring according to the Wedderburn little theorem [3].…”

Section: Operationsmentioning

confidence: 99%

Predictable universally unique identification of sequential events on complex objects

Pereira-Santos,

Dalforno,

Carvalho

2021

Preprint

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Universal identifiers and hashing have been widely adopted in computer science from distributed financial transactions to data science. This is a consequence of their capability to avoid many shortcomings of relative identifiers, such as limited scope and the need for central management. However, the current identifiers in use are isolated entities which cannot provide much information about the relationship between objects. As an example, if one has both the identifiers of an object and its transformed version, no information about how they are related can be obtained without resorting to an authority or additionally appended information. Moreover, given an input object and an arbitrarily long sequence of costly steps, one cannot currently predict the identifier of the outcome without actually processing the entire sequence. The capability of predicting the resulting identifier and avoiding redundant calculations is highly desirable in an efficient unmanaged system. In this paper, we propose a new kind of unique identifier which is calculated from the list of events that can produce an object, instead of directly identifying it by content. This provides an identification scheme regardless of the object's current existence, thus allowing to inexpensively check for its content in a database and retrieve it when it has already been calculated before. We propose these identifiers in the context of abstract algebra, where objects are represented by elements that can be operated according to useful properties, such as associativity, order sensitivity when desired, and reversibility, while simplicity of implementation and strong guarantees are given by well-known group theory results.

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Completing Segre's proof of Wedderburn's little theorem

Cited by 6 publications

References 21 publications

Variations on a Theme of Glauberman

Variations on a Theme of Glauberman

Two cryptomorphic formalizations of projective incidence geometry

Predictable universally unique identification of sequential events on complex objects

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