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Journal articles on the topic 'Linear algebraic groups over finite fields'

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Published: 4 June 2021

Last updated: 28 July 2025

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1

BASHKIROV, E. L., and C. K. GUPTA. "LINEAR GROUPS OVER LOCALLY FINITE EXTENSIONS OF INFINITE FIELDS." International Journal of Algebra and Computation 17, no. 05n06 (2007): 905–22. http://dx.doi.org/10.1142/s0218196707003937.

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Let P be a field of characteristic different from 2, let K be an associative commutative P-algebra with an identity 1 and let n be an integer, n ≥ 2. Assume that K is an algebraic extension of P having, in general, zero divisors and P is an algebraic separable extension of an infinite subfield k. The paper studies subgroups X of the group GLn (K) such that X contains a root k-subgroup, i.e. a subgroup which is conjugate in GLn (K) to a group of all matrices [Formula: see text], a ∈ k.

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2

Detinko, A. S., and D. L. Flannery. "NILPOTENT PRIMITIVE LINEAR GROUPS OVER FINITE FIELDS." Communications in Algebra 33, no. 2 (2005): 497–505. http://dx.doi.org/10.1081/agb-200047422.

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3

Altinel, Tuna, and Gregory Cherlin. "On central extensions of algebraic groups." Journal of Symbolic Logic 64, no. 1 (1999): 68–74. http://dx.doi.org/10.2307/2586751.

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In this paper the following theorem is proved regarding groups of finite Morley rank which are perfect central extensions of quasisimple algebraic groups.Theorem 1. Let G be a perfect group of finite Morley rank and let C0be a definable central subgroup of G such that G/C0 is a universal linear algebraic group over an algebraically closed field; that is G is a perfect central extension of finite Morley rank of a universal linear algebraic group. Then C0 = 1.Contrary to an impression which exists in some circles, the center of the universal extension of a simple algebraic group, as an abstract

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4

Rossmann, Tobias. "Primitive finite nilpotent linear groups over number fields." Journal of Algebra 451 (April 2016): 248–67. http://dx.doi.org/10.1016/j.jalgebra.2015.11.044.

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5

Wood, Jay A. "Isometry groups of additive codes over finite fields." Journal of Algebra and Its Applications 17, no. 10 (2018): 1850198. http://dx.doi.org/10.1142/s0219498818501980.

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When [Formula: see text] is a linear code over a finite field [Formula: see text], every linear Hamming isometry of [Formula: see text] to itself is the restriction of a linear Hamming isometry of [Formula: see text] to itself, i.e. a monomial transformation. This is no longer the case for additive codes over non-prime fields. Every monomial transformation mapping [Formula: see text] to itself is an additive Hamming isometry, but there may exist additive Hamming isometries that are not monomial transformations.The monomial transformations mapping [Formula: see text] to itself form a group [For

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6

Bezushchak, Oksana. "Automorphisms of Mackey groups." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 2 (2023): 16–19. http://dx.doi.org/10.17721/1812-5409.2023/2.2.

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We consider total subspaces of linear functionals on an infinite-dimensional vector space and the related Mackey algebras and groups. We outline the description of automorphisms of Mackey groups SL∞(V|W), O∞(f), and SU∞(f) over fields of characteristics not equal to 2, 3. Moreover, the paper explores the relationship between field automorphisms and automorphisms of the aforementioned groups. J.Hall proved that infinite simple finitary torsion groups are the alternating groups on infinite sets or Mackey groups over a field, which is an algebraic extension of a finite field. J.Schreier and S.Ula

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7

Waterhouse, William C. "Two generators for the general linear groups over finite fields." Linear and Multilinear Algebra 24, no. 4 (1989): 227–30. http://dx.doi.org/10.1080/03081088908817916.

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8

Blau, H. I., and J. P. Zhang. "Linear Groups of Small Degree over Fields of Finite Characteristic." Journal of Algebra 159, no. 2 (1993): 358–86. http://dx.doi.org/10.1006/jabr.1993.1162.

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9

Ghaani Farashahi, Arash. "Wave Packet Transform over Finite Fields." Electronic Journal of Linear Algebra 30 (February 8, 2015): 507–29. http://dx.doi.org/10.13001/1081-3810.2903.

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In this article we introduce the notion of finite wave packet groups over finite fields as the finite group of dilations, translations, and modulations. Then we will present a unified theoretical linear algebra approach to the theory of wave packet transform (WPT) over finite fields. It is shown that each vector defined over a finite field can be represented as a coherent sum of finite wave packet group elements as well.

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10

Letellier, Emmanuel. "Tensor products of unipotent characters of general linear groups over finite fields." Transformation Groups 18, no. 1 (2013): 233–62. http://dx.doi.org/10.1007/s00031-013-9211-3.

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11

Chen, Yin, and Xinxin Zhang. "A Class of Quadratic Matrix Equations over Finite Fields." Algebra Colloquium 30, no. 01 (2022): 169–80. http://dx.doi.org/10.1142/s1005386723000147.

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We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials. We also find a generating set for the vanishing ideal of these orbits.

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12

WEHRFRITZ, B. A. F. "FINITE-FINITARY GROUPS OF AUTOMORPHISMS." Journal of Algebra and Its Applications 01, no. 04 (2002): 375–89. http://dx.doi.org/10.1142/s0219498802000318.

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In this paper we attempt to describe the structure of groups G of automorphisms of an abelian group M with the property that M(g - 1) is finite for every element g of G. These groups are closely related to the finitary linear groups over finite fields. The abelian case is critical for our work and the core result of this paper is the following. An abelian group A is isomorphic to a group G as above with M torsion if and only if A is torsion and has a residually-finite subgroup B with A/B a direct sum of cyclic groups.

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13

Donkin, Stephen, and Karin Erdmann. "James Alexander Green. 26 February 1926—7 April 2014." Biographical Memoirs of Fellows of the Royal Society 67 (August 14, 2019): 173–90. http://dx.doi.org/10.1098/rsbm.2019.0012.

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James Alexander Green, known as Sandy, was a mathematician of great influence and distinction. He was an algebraist, famous for his work on modular representations of finite groups, and the development of the theory of polynomial representations of general linear groups. He was elected Fellow of the Royal Society of Edinburgh (1968) and Fellow of the Royal Society of London (1987). He was awarded prizes of the London Mathematical Society, a Senior Berwick Prize (in 1984) and the De Morgan Medal (in 2001). In his doctoral thesis, on semigroups, Sandy introduced fundamental relations, now known

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14

Nasybullov, Timur. "Reidemeister spectrum of special and general linear groups over some fields contains 1." Journal of Algebra and Its Applications 18, no. 08 (2019): 1950153. http://dx.doi.org/10.1142/s0219498819501536.

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We prove that if [Formula: see text] is an algebraically closed field of zero characteristic which has infinite transcendence degree over [Formula: see text], then there exists a field automorphism [Formula: see text] of [Formula: see text] and [Formula: see text] such that [Formula: see text]. This fact implies that [Formula: see text] and [Formula: see text] do not possess the [Formula: see text]-property. However, if the transcendece degree of [Formula: see text] over [Formula: see text] is finite, then [Formula: see text] and [Formula: see text] are known to possess the [Formula: see text]

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15

Vincent, R., and A. E. Zalesski. "Non-Hurwitz Classical Groups." LMS Journal of Computation and Mathematics 10 (2007): 21–82. http://dx.doi.org/10.1112/s1461157000001303.

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AbstractIn previous work by Di Martino, Tamburini and Zalesski [Comm. Algebra28 (2000) 5383–5404] it is shown that certain low-dimensional classical groups over finite fields are not Hurwitz. In this paper the list is extended by adding the special linear and special unitary groups in dimensions 8.9,11.13. We also show that all groups Sp(n, q) are not Hurwitz forqeven andn= 6,8,12,16. In the range 11 <n< 32 many of these groups are shown to be non-Hurwitz. In addition, we observe that PSp(6, 3),PΩ±(8, 3k),PΩ±10k), Ω(11,3k), Ω±(14,3k), Ω±(16,7k), Ω(n, 7k) forn= 9,11,13, PSp(8, 7k) are not

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16

Lo, Catharine Wing Kwan, та Matilde Marcolli. "𝔽ζ-geometry, Tate motives, and the Habiro ring". International Journal of Number Theory 11, № 02 (2015): 311–39. http://dx.doi.org/10.1142/s1793042115500189.

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In this paper, we propose different notions of 𝔽ζ-geometry, for ζ a root of unity, generalizing notions of 𝔽1-geometry (geometry over the "field with one element") based on the behavior of the counting functions of points over finite fields, the Grothendieck class, and the notion of torification. We relate 𝔽ζ-geometry to formal roots of Tate motives, and to functions in the Habiro ring, seen as counting functions of certain ind-varieties. We investigate the existence of 𝔽ζ-structures in examples arising from general linear groups, matrix equations over finite fields, and some quantum modular f

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17

Mérai, László. "Values of rational functions in small subgroups of finite fields and the identity testing problem from powers." International Journal of Number Theory 16, no. 02 (2019): 219–31. http://dx.doi.org/10.1142/s1793042120500128.

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Motivated by some algorithmic problems, we give lower bounds on the size of the multiplicative groups containing rational function images of low-dimensional affine subspaces of a finite field [Formula: see text] considered as a linear space over a subfield [Formula: see text]. We apply this to the recently introduced algorithmic problem of identity testing of “hidden” polynomials [Formula: see text] and [Formula: see text] over a high degree extension of a finite field, given oracle access to [Formula: see text] and [Formula: see text].

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18

BARTELS, H. J., and D. A. MALININ. "ON FINITE GALOIS STABLE SUBGROUPS OF GLn IN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS." Journal of Algebra and Its Applications 08, no. 04 (2009): 493–503. http://dx.doi.org/10.1142/s0219498809003400.

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Let K/ℚ be a finite Galois extension with maximal order [Formula: see text] and Galois group Γ. For finite Γ-stable subgroups [Formula: see text] it is known [4], that they are generated by matrices with coefficients in [Formula: see text], Kab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more p

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19

Xing, Chaoping, and Sze Ling Yeo. "New Linear Codes and Algebraic Function Fields Over Finite Fields." IEEE Transactions on Information Theory 53, no. 12 (2007): 4822–25. http://dx.doi.org/10.1109/tit.2007.909125.

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20

Scanlon, Thomas, and José Felipe Voloch. "Difference algebraic subgroups of commutative algebraic groups over finite fields." manuscripta mathematica 99, no. 3 (1999): 329–39. http://dx.doi.org/10.1007/s002290050176.

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21

Behr, Helmut. "Finite presentability of arithmetic groups over global function fields." Proceedings of the Edinburgh Mathematical Society 30, no. 1 (1987): 23–39. http://dx.doi.org/10.1017/s0013091500017934.

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Arithmetic subgroups of reductive algebraic groups over number fields are finitely presentable, but over global function fields this is not always true. All known exceptions are “small” groups, which means that either the rank of the algebraic group or the set S of the underlying S-arithmetic ring has to be small. There exists now a complete list of all such groups which are not finitely generated, whereas we onlyhave a conjecture which groups are finitely generated but not finitely presented.

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22

Zhang, Ji Ping. "On linear groups over finite fields." Proceedings of the American Mathematical Society 110, no. 1 (1990): 53. http://dx.doi.org/10.1090/s0002-9939-1990-1028297-1.

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23

Dong, Xue Dong. "Linear Block Codes for Six-Dimensional Signals over Finite Fields." Applied Mechanics and Materials 385-386 (August 2013): 1358–61. http://dx.doi.org/10.4028/www.scientific.net/amm.385-386.1358.

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t is known that the performance of a signal constellation used to transmit digital information over the additive white Gaussian noise channel can be improved by increasing the dimensionality of the signal set used for transmission. This paper derives an algorithm for constructing codes for six-dimensional signals over finite fields of the algebraic integer ring of the cyclotomic field modulo irreducible elements with the norm , where is a prime number and or .These linear codes can correct some types of errors and provide an algebraic approach in an area which is currently mainly dominated by

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24

Elmer, Jonathan, and Martin Kohls. "Zero-Separating Invariants for Linear Algebraic Groups." Proceedings of the Edinburgh Mathematical Society 59, no. 4 (2015): 911–24. http://dx.doi.org/10.1017/s0013091515000322.

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AbstractAbstract Let G be a linear algebraic group over an algebraically closed field 𝕜 acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and , respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL2(𝕜) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero,

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25

Conrad, Brian. "Finiteness theorems for algebraic groups over function fields." Compositio Mathematica 148, no. 2 (2011): 555–639. http://dx.doi.org/10.1112/s0010437x11005665.

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AbstractWe prove the finiteness of class numbers and Tate–Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semi-simple groups via systematic use of the recently developed structure theory and classification of pseudo-reductive groups.

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26

Li, Ying. "On Finite Subgroups in the General Linear Groups over an Algebraic Number Field." Journal of Physics: Conference Series 2287, no. 1 (2022): 012006. http://dx.doi.org/10.1088/1742-6596/2287/1/012006.

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Abstract As is well-known, there are only finitely many isomorphic classes of finite subgroups in a given general linear group over the field of rational numbers. This result can be generalized to any algebraic number field. While the case of field of rational numbers is relatively well-studied, we still do not know much for general algebraic number field cases. In this article, we discuss the finiteness of isomorphic classes of finite subgroups of general linear groups over an algebraic number field. We give a method to calculate a multiplicative bound for the orders of finite subgroups and t

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27

Wagner, Frank. "Fields of finite Morley rank." Journal of Symbolic Logic 66, no. 2 (2001): 703–6. http://dx.doi.org/10.2307/2695038.

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AbstractIf K is a field of finite Morley rank, then for any parameter set A ⊆ Keq the prime model over A is equal to the model-theoretic algebraic closure of A. A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl(∅).

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28

Bhowmick, Sanjit, Deepak Kumar Dalai, and Sihem Mesnager. "On linear complementary pairs of algebraic geometry codes over finite fields." Discrete Mathematics 347, no. 12 (2024): 114193. http://dx.doi.org/10.1016/j.disc.2024.114193.

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29

Rossmann, Tobias. "Primitivity testing of finite nilpotent linear groups." LMS Journal of Computation and Mathematics 14 (March 1, 2011): 87–98. http://dx.doi.org/10.1112/s1461157010000227.

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AbstractWe describe a practical algorithm for primitivity testing of finite nilpotent linear groups over various fields of characteristic zero, including number fields and rational function fields over number fields. For an imprimitive group, a system of imprimitivity can be constructed. An implementation of the algorithm in Magma is publicly available.

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30

Praeger, Cheryl E. "Kronecker classes of fields and covering subgroups of finite groups." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 57, no. 1 (1994): 17–34. http://dx.doi.org/10.1017/s1446788700036028.

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AbstractKronecker classes of algebraci number fields were introduced by W. Jehne in an attempt to understand the extent to which the structure of an extension K: k of algebraic number fields was influenced by the decomposition of primes of k over K. He found an important link between Kronecker equivalent field extensions and a certain covering property of their Galois groups. This surveys recent contributions of Group Theory to the understanding of Kronecker equivalence of algebraic number fields. In particular some group theoretic conjectures related to the Kronecker class of an extension of

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31

Cook, Brian, and Ákos Magyar. "On restricted arithmetic progressions over finite fields." Online Journal of Analytic Combinatorics, no. 7 (December 31, 2012): 1–10. https://doi.org/10.61091/ojac-701.

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Let \( A \) be a subset of \( \mathbb{F}_p^n \), the \( n \)-dimensional linear space over the prime field \( \mathbb{F}_p \), of size at least \( \delta N \) (\( N = p^n \)), and let \( S_v = P^{-1}(v) \) be the level set of a homogeneous polynomial map \( P : \mathbb{F}_p^n \to \mathbb{F}_p^R \) of degree \( d \), for \( v \in \mathbb{F}_p^R \). We show that, under appropriate conditions, the set \( A \) contains at least \( c N|S| \) arithmetic progressions of length \( l \leq d \) with common difference in \( S_v \), where \( c \) is a positive constant depending on \( \delta \), \( l \),

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32

Köroğlu, Mehmet Emin, and Mustafa Sarı. "Skew Constacyclic Codes over a Non-Chain Ring." Entropy 25, no. 3 (2023): 525. http://dx.doi.org/10.3390/e25030525.

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In this paper, we investigate the algebraic structure of the non-local ring Rq=Fq[v]/⟨v2+1⟩ and identify the automorphisms of this ring to study the algebraic structure of the skew constacyclic codes and their duals over this ring. Furthermore, we give a necessary and sufficient condition for the skew constacyclic codes over Rq to be linear complementary dual (LCD). We present some examples of Euclidean LCD codes over Rq and tabulate the parameters of Euclidean LCD codes over finite fields as the Φ-images of these codes over Rq, which are almost maximum distance separable (MDS) and near MDS. E

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33

Minchenko, Andrey, Alexey Ovchinnikov, and Michael F. Singer. "Unipotent differential algebraic groups as parameterized differential Galois groups." Journal of the Institute of Mathematics of Jussieu 13, no. 4 (2013): 671–700. http://dx.doi.org/10.1017/s1474748013000200.

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AbstractWe deal with aspects of direct and inverse problems in parameterized Picard–Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) $G$ is a PPV Galois group over these fields if and only if $G$ contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs $G$, including unipotent groups, $G$ is such a group if and only if it has differential type $0$. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois

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34

FLANNERY, D. L., and E. A. O'BRIEN. "LINEAR GROUPS OF SMALL DEGREE OVER FINITE FIELDS." International Journal of Algebra and Computation 15, no. 03 (2005): 467–502. http://dx.doi.org/10.1142/s0218196705002426.

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For n = 2,3 and finite field 𝔼 of characteristic greater than n, we provide a complete and irredundant list of soluble irreducible subgroups of GL (n,𝔼). The insoluble irreducible subgroups of GL (2,𝔼) are similarly determined. Each group is given explicitly by a generating set of matrices. The lists are available electronically.

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35

Parimala, R., P. Gille, and J. L. Colliot-Th�l�ne. "Arithmetic of linear algebraic groups over 2-dimensional geometric fields." Duke Mathematical Journal 121, no. 2 (2004): 285–341. http://dx.doi.org/10.1215/s0012-7094-04-12124-4.

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36

Müller, Peter. "Algebraic groups over finite fields, a quick proof of Lang’s theorem." Proceedings of the American Mathematical Society 131, no. 2 (2002): 369–70. http://dx.doi.org/10.1090/s0002-9939-02-06591-7.

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37

Perret-Gentil, Corentin. "Exponential Sums Over Finite Fields and the Large Sieve." International Mathematics Research Notices 2020, no. 20 (2018): 7139–74. http://dx.doi.org/10.1093/imrn/rny202.

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Abstract By using a variant of the large sieve for Frobenius in compatible systems developed in [24] and [27], we obtain zero-density estimates for arguments of $\ell $-adic trace functions over finite fields with values in some algebraic subsets of the cyclotomic integers, when the monodromy groups are known. This applies in particular to hyper-Kloosterman sums and general exponential sums considered by Katz.

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38

Malinin, Dmitry. "On the integral and globally irreducible representations of finite groups." Journal of Algebra and Its Applications 17, no. 05 (2018): 1850087. http://dx.doi.org/10.1142/s0219498818500871.

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We consider the arithmetic of integral representations of finite groups over algebraic integers and the generalization of globally irreducible representations introduced by Van Oystaeyen and Zalesskii. For the ring of integers [Formula: see text] of an algebraic number field [Formula: see text] we are interested in the question: what are the conditions for subgroups [Formula: see text] such that [Formula: see text], the [Formula: see text]-span of [Formula: see text], coincides with [Formula: see text], the ring of [Formula: see text]-matrices over [Formula: see text], and what are the minimal

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39

Moldovyan, Alexandr, and Nikolay Moldovyan. "Vector finite fields of characteristic two as algebraic support of multivariate cryptography." Computer Science Journal of Moldova 32, no. 1(94) (2024): 46–60. http://dx.doi.org/10.56415/csjm.v32.04.

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The central issue of the development of the multivariate public key algorithms is the design of reversible non-linear mappings of $n$-dimensional vectors over a finite field, which can be represented in a form of a set of power polynomials. For the first time, finite fields $GF\left((2^d)^m\right)$ of characteristic two, represented in the form of $m$-dimensional finite algebras over the fields $GF(2^d)$ are introduced for implementing the said mappings as exponentiation operation. This technique allows one to eliminate the use of masking linear mappings, usually used in the known approaches t

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40

Cohen, S. D. "The factorable core of polynomials over finite fields." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 49, no. 2 (1990): 309–18. http://dx.doi.org/10.1017/s1446788700030585.

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AbstractFor a polynomial f(x) over a finite field Fq, denote the polynomial f(y)−f(x) by ϕf(x, y). The polynomial ϕf has frequently been used in questions on the values of f. The existence is proved here of a polynomial F over Fq of the form F = Lr, where L is an affine linearized polynomial over Fq, such that f = g(F) for some polynomial g and the part of ϕf which splits completely into linear factors over the algebraic closure of Fq is exactly φF. This illuminates an aspect of work of D. R. Hayes and Daqing Wan on the existence of permutation polynomials of even degree. Related results on va

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41

Jitman, Somphong. "Linear intersection pairs of negacyclic codes and their applications in construction of entanglement-assisted quantum error-correcting codes." Journal of Physics: Conference Series 2793, no. 1 (2024): 012012. http://dx.doi.org/10.1088/1742-6596/2793/1/012012.

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Abstract Linear intersection pairs of linear codes have become of interest and continuously studied due to their nice algebraic properties and wide applications. In this article, we focus on linear intersection pairs of negacyclic codes over finite fields and their applications. General characterization and algebraic properties of such pairs are given in terms of their generator polynomials. For s∈{0,1}, explicit constructions of linear s-intersection pairs and linear s-complementary pairs of negacyclic codes are presented. As applications, constructions of entanglement-assisted quantum error-

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42

Stulemeijer, Thierry. "CHABAUTY LIMITS OF ALGEBRAIC GROUPS ACTING ON TREES THE QUASI-SPLIT CASE." Journal of the Institute of Mathematics of Jussieu 19, no. 4 (2018): 1031–91. http://dx.doi.org/10.1017/s1474748018000282.

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Given a locally finite leafless tree $T$, various algebraic groups over local fields might appear as closed subgroups of $\operatorname{Aut}(T)$. We show that the set of closed cocompact subgroups of $\operatorname{Aut}(T)$ that are isomorphic to a quasi-split simple algebraic group is a closed subset of the Chabauty space of $\operatorname{Aut}(T)$. This is done via a study of the integral Bruhat–Tits model of $\operatorname{SL}_{2}$ and $\operatorname{SU}_{3}^{L/K}$, that we carry on over arbitrary local fields, without any restriction on the (residue) characteristic. In particular, we show

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DETINKO, A. S., and D. L. FLANNERY. "CLASSIFICATION OF NILPOTENT PRIMITIVE LINEAR GROUPS OVER FINITE FIELDS." Glasgow Mathematical Journal 46, no. 3 (2004): 585–94. http://dx.doi.org/10.1017/s0017089504002046.

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44

KHREBTOVA, EKATERINA S., and DMITRY MALININ. "ON FINITE GALOIS STABLE ARITHMETIC GROUPS AND THEIR APPLICATIONS." Journal of Algebra and Its Applications 07, no. 06 (2008): 773–83. http://dx.doi.org/10.1142/s0219498808003119.

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We prove the existence and finiteness theorems for integral representations stable under Galois operation. An explicit construction of the realization fields for representations of finite groups stable under the natural operation of the Galois group is given. We also compare the representations over fields and the rings of integers, and give a quantitative result on the rarity of integral Galois stable representations. There is a series of related conjectures and applications to arithmetic algebraic geometry, finite flat group schemes, positive definite quadratic lattices and Galois cohomology

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Morita, Jun, and Bertrand Rémy. "Simplicity of Some Twin Tree Automorphism Groups with Trivial Commutation Relations." Canadian Mathematical Bulletin 57, no. 2 (2014): 390–400. http://dx.doi.org/10.4153/cmb-2014-002-2.

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AbstractWe prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).

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COHEN, BOAZ. "MULTIPLICATIVE n-TH ROOT FUNCTIONS OVER FINITE SEMIGROUPS, GROUPS, FIELDS AND COMMUTATIVE RINGS." Mathematical Reports 26(76), no. 3-4 (2024): 219–50. https://doi.org/10.59277/mrar.2024.26.76.3.4.219.

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In this paper, we study the existence and uniqueness of multiplicative n-th root functions n √ over finite semigroups, in order to implement these ideas on finite groups, fields and commutative rings. A set of sufficient and necessary conditions are presented for existence of multiplicative n-th root functions over different algebraic structures. It is also shown that once the existence is established, the uniqueness is guaranteed. In addition, we describe the construction procedure of such a function.

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PELLEGRINI, MARCO ANTONIO. "THE -GENERATION OF THE SPECIAL LINEAR GROUPS OVER FINITE FIELDS." Bulletin of the Australian Mathematical Society 95, no. 1 (2016): 48–53. http://dx.doi.org/10.1017/s0004972716000617.

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We complete the classification of the finite special linear groups $\text{SL}_{n}(q)$ which are $(2,3)$-generated, that is, which are generated by an involution and an element of order $3$. This also gives the classification of the finite simple groups $\text{PSL}_{n}(q)$ which are $(2,3)$-generated.

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Shlepkin, A. A. "On Locally Finite Groups Saturated by Full and Special Linear Groups over Finite Fields." Siberian Mathematical Journal 65, no. 4 (2024): 869–77. http://dx.doi.org/10.1134/s003744662404013x.

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Minaiev, P. Ye, O. O. Pypka, and I. V. Shyshenko. "On Poisson (2-3)-algebras which are finite-dimensional over the center." Researches in Mathematics 32, no. 1 (2024): 118. http://dx.doi.org/10.15421/242411.

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One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/\zeta(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $n$-groups, associative algebras, Lie algebras, Lie $n$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Sc

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Borovoi, Mikhail, Boris Kunyavskiı̆, and Philippe Gille. "Arithmetical birational invariants of linear algebraic groups over two-dimensional geometric fields." Journal of Algebra 276, no. 1 (2004): 292–339. http://dx.doi.org/10.1016/j.jalgebra.2003.10.024.

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