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Expressing the statement of the Feit-Thompson theorem with diagrams in the category of finite groups
Gavrilovich M.
We reformulate the statement of the Feit-Thompson theorem in terms of diagrams in the category of finite groups, namely iterations of the Quillen lifting property with respect to particular morphisms.
Publication based on the results of:
Batanin M., White D., Journal of Pure and Applied Algebra 2024 Vol. 228 No. 6 Article 107570
Given a combinatorial (semi-)model category M and a set of morphisms C, we establish the existence of a semi-model category LCM satisfying the universal property of the left Bousfield localization in the category of semi-model categories. Our main tool is a semi-model categorical version of a result of Jeff Smith, that appears to be of ...
Added: December 26, 2025
Arzhantsev I., Petravchuk A. P., Mathematical notes 2009 Vol. 86 No. 5 P. 625–628
Every subfield K(φ) of the field of rational fractions K(x_1, . . . , x_n) is contained in a unique maximal subfield of the form K(ψ). The element ψ is said to be generating for the element φ. A subfield of K(x_1, . . . , x_n) is said to be saturated if, together with ...
Added: June 13, 2025
Farjoun E. D., Ivanov S. O., Krasilnikov A. et al., Journal of Algebra 2025 Vol. 675 P. 273–288
We consider several types of non-existence theorems for functors. For example, there are no nontrivial functors from the category of groups (or the category of pointed sets, or vector spaces) to any small category. In addition, we consider questions about nonexistence of subfunctors and quotients of the identity functor on the category of groups (or ...
Added: May 5, 2025
Prokhorov Y., , in: 8th European Congress of Mathematics, Portorož, 20–26 June, 2021.: [б.и.], 2023. P. 413–437.
We survey new results on finite groups of birational transformations of algebraic varieties. ...
Added: November 13, 2023
Raikov A., Pirani M., IEEE Access 2022 Vol. 10 P. 56296–56315
The goal of the paper is to find means for the unification of human-machine duality in collective behavior of people and machines, by conciliating approaches that proceed in opposite directions. The first approach proceeds top-down from non-formalizable, cognitive, uncaused, and chaotic human consciousness towards purposeful and sustainable human-machine interaction. The second approach proceeds bottom-up from ...
Added: July 19, 2022
Prokhorov Y., / Series arXiv "math". 2021.
We survey new results on finite groups of birational transformations of algebraic varieties. ...
Added: November 25, 2021
Арутюнов А. А., Kosolapov L., Finite Fields and Their Applications 2021 Vol. 76 Article 101921
In this paper we establish decomposition theorems for derivations of group rings. We provide a topological technique for studying derivations of a group ring A[G] in case G has finite conjugacy classes. As a result, we describe all derivations of algebra A[G] for the case when G is a finite group, or G is an FC-group. In addition, we describe an algorithm to explicitly calculate all derivations of ...
Added: October 4, 2021
Shramov K., European Journal of Mathematics 2021 Vol. 7 P. 591–612
We classify finite groups that can act by automorphisms and birational automorphisms on non-trivial Severi–Brauer surfaces over fields of characteristic zero. ...
Added: September 8, 2021
Rumynin D., Taylor J., Linear Algebra and its Applications 2021 Vol. 610 P. 135–168
We use the structure of finite-dimensional graded algebras to develop the theory of antilinear representations of finite C_2-graded groups. A finite C_2-graded group is a finite group with a subgroup of index 2. In this theory the subgroup acts linearly, while the other coset acts antilinearly. We introduce antilinear blocks, whose structure is a crucial ...
Added: September 7, 2021
Shramov K., / Series arXiv "math". 2020.
We classify finite groups that can act by automorphisms and birational automorphisms on non-trivial Severi-Brauer surfaces over fields of characteristic zero. ...
Added: August 19, 2020
Shramov K., / Series arXiv "math". 2019.
We show that automorphism groups of Hopf and Kodaira surfaces have unbounded finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces, we make some observations on finite groups acting along the fibers and on the base of such a fibration. ...
Added: November 19, 2019
Popov V. L., Doklady Mathematics 2018 Vol. 98 No. 2 P. 413–415
The compressibility of certain types of finite groups of birational automorphisms of algebraic varieties is established. ...
Added: November 13, 2018
Rodin A., , in: Philosophy, Mathematics, Linguistics: Aspects of Interaction.: [б.и.], 2009. P. 171–175.
A perpetual change of foundations observed in the real history of the discipline is not a
historical accident but an essential feature of foundations. I distinguish between the progress of
mathematics and renewal of its foundations and show how the latter contributes to the former with
some historical examples. I also describe a mechanism of renewal of foundations, ...
Added: June 7, 2018
Gavrilovich M., / Series arxiv "math.CT". 2017.
We illustrate the generative power of the lifting property (orthogonality of morphisms in a category) as means of defining natural elementary mathematical concepts by giving a number of examples in various categories, in particular showing that many standard elementary notions of abstract topology can be defined by applying the lifting property to simple morphisms of ...
Added: July 21, 2017
Gavrilovich M., The De Morgan Gazette 2014 Vol. 5 No. 4 P. 23–32
We observe that some natural mathematical definitions are lifting properties relative to simplest counterexamples, namely the definitions of surjectivity and injectivity of maps, as well as of being connected, separation axioms T0 and T1 in topology, having dense image, induced (pullback) topology, and every real-valued function being bounded (on a connected domain). We also offer ...
Added: October 20, 2015
Balzin E., Успехи математических наук 2014 Т. 69 № 5(419) С. 159–160
В статье дан обзор части результатов диссертационной работы автора. Речь идет о применении идеи категорного разрешения сингулярностей, которая была активно опробована алгебраическими геометрами для триангулированных категорий, в гомотопической алгебре. В связи с тем, что возникающие тут категории не имеют никакой аддитивной структуры, возникает необходимость в разработке новых методов. В рамках формализма Сигала, который позволяет описывать ...
Added: December 24, 2014
Balzin Edouard, / Series math "arxiv.org". 2014.
In the world of triangulated categories, categorical resolutions (as defined by Kuznetsov and Luntz) have been useful. One would like to have a similar notion of categorical resolution in homotopical algebra, where one works with categories which are not additive, such as the categories of E_n-algebras. Describing algebraic structures using the approach inspired by Segal, ...
Added: December 23, 2014
Positselski L., / Series math "arxiv.org". 2014. No. 1404.5011.
We construct the reduction of an exact category with a twist functor with respect to an element of its graded center in presence of an exact-conservative forgetful functor annihilating this central element. The procedure allows, e.g., to recover the abelian/exact category of modular representations of a finite group from the exact category of its l-adic ...
Added: April 22, 2014
Busjatskaja I., Monastyrsky M. I., , in: Symmetry: Art and Science. Special Issue for the Congress-Festival of ISIS-Symmetry.: Athens: The Hellenic Open University, 2013. P. 248–253.
The paper examines the significance of Platonic Solids in different parts of modern science. ...
Added: January 30, 2014
Athens: The Hellenic Open University, 2013.
The book contains the reports of the member of the congress from the different countres. They consider the idea of the symmety in the science and in the art. ...
Added: January 30, 2014