Cycle indices of linear, affine, and projective groups

In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general linear group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.

Grassmann codes are linear codes associated with the Grassmann variety $G(\ell,m)$ of $\ell$-dimensional subspaces of an $m$ dimensional vector space $\mathbb{F}_{q}^{m}.$ They were studied by Nogin for general $q.$ These codes are conveniently described using the correspondence between non-degenerate $[n,k]_{q}$ linear codes on one hand and non-degenerate $[n,k]$ projective systems on the other hand. A non-degenerate $[n,k]$ projective system is simply a collection of $n$ points in projective space $\mathbb{P}^{k-1}$ satisfying the condition that no hyperplane of $\mathbb{P}^{k-1}$ contains all the $n$ points under consideration. In this paper we will determine the weight of linear codes $C(3,8)$ associated with Grassmann varieties $G(3,8)$ over an arbitrary finite field $\mathbb{F}_{q}$. We use a formula for the weight of a codeword of $C(3,8)$, in terms of the cardinalities certain varieties associated with alternating trilinear forms on $\mathbb{F}_{q}^{8}.$ For $m=6$ and $7,$ the weight spectrum of $C(3,m)$ associated with $G(3,m),$ have been fully determined by Kaipa K.V, Pillai H.K and Nogin Y. A classification of trivectors depends essentially on the dimension $n$ of the base space. For $n\leq 8 $ there exist only finitely many trivector classes under the action of the general linear group $GL(n).$ The methods of Galois cohomology can be used to determine the classes of nondegenerate trivectors which split into multiple classes when going from $\mathbb{\bar{F}}$ to $\mathbb{F}.$ This program is partially determined by Noui L. and Midoune N. and the classification of trilinear alternating forms on a vector space of dimension $8$ over a finite field $\mathbb{F}_{q}$ of characteristic other than $2$ and $3$ was solved by Noui L. and Midoune N. We describe the $\mathbb{F}_{q}$-forms of $2$-step splitting trivectors of rank $8$, where char $\mathbb{F}_{q}\neq 3.$ This fact we use to determine the weight of the $\mathbb{F}_{q}$-forms.

The finite sharply $2$-transitive groups were classified by Zassenhaus in the 1930's. They essentially all look like the group of affine linear transformations $x\mapsto ax+b$ for some field (or at least near-field) $K$. However, the question remained open whether the same is true for infinite sharply $2$-transitive groups. There has been extensive work on the structures associated to such groups indicating that Zassenhaus' results might extend might extend to the infinite setting. For many specific classes of groups, like Lie groups, linear groups, or groups definable in o-minimal it was indeed proved that all examples inside the given class arise in this way as affine groups. However, it recently turned out that the reason for the lack of a general proof was the fact that there are plenty of sharply $2$-transitive groups which do not arise from fields or near-fields! In fact, it is not too hard to construct concrete examples (see below). In this note, we survey general sharply $n$-transitive groups and describe how to construct examples not arising from fields.

Let f̃(X) ∈ ℤ[X] be a degree-n polynomial such that f(X): = f̃(X)od p factorizes into n distinct linear factors over 𝔽_p. We study the problem of deterministically factoring f(X) over 𝔽_p given f̃(X). Under the generalized Riemann hypothesis (GRH), we give an improved deterministic algorithm that computes the complete factorization of f(X) in the case that the Galois group of f̃(X) is (permutation isomorphic to) a linear group G ≤ GL(V) on the set S of roots of f̃(X), where V is a finite-dimensional vector space over a finite field 𝔽 and S is identified with a subset of V. In particular, when |S| = |V|^{Ω(1)}, the algorithm runs in time polynomial in n^{log n/(log log log log n)^{1/3}} and the size of the input, improving Evdokimov’s algorithm. Our result also applies to a general Galois group G when combined with a recent algorithm of the author. 
\nTo prove our main result, we introduce a family of objects called linear m-schemes and reduce the problem of factoring f(X) to a combinatorial problem about these objects. We then apply techniques from additive combinatorics to obtain an improved bound. Our techniques may be of independent interest.

Previous article Next article On the Classification of Boolean Functions by the General Linear and Affine GroupsMichael A. HarrisonMichael A. Harrisonhttps://doi.org/10.1137/0112026PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] E. Artin, Geometric algebra, Interscience Publishers, Inc., New York-London, 1957x+214 MR0082463 (18,553e) 0077.02101 Google Scholar[2] Robert L. Ashenhurst, The application of counting techniques, Proceedings of the Association for Computing Machinery, Pittsburgh, 1952, Richard Rimbach Associates, Pittsburgh, Pa., 1952, 293–305 MR0056562 (15,93h) Google Scholar[3] Randolph Church, Tables of irreducible polynomials for the first four prime moduli, Ann. of Math. (2), 36 (1935), 198–209 MR1503219 0011.00501 CrossrefGoogle Scholar[4] Leonard Eugene Dickson, Linear groups: With an exposition of the Galois field theory, with an introduction by W. Magnus, Dover Publications Inc., New York, 1958xvi+312 MR0104735 (21:3488) 0082.24901 Google Scholar[5] B. Elspas, Autonomous linear sequential networks, I. R. E. Trans., CT-6 (1959), 45–60, March Google Scholar[6] Michael A. Harrison, The number of transitivity sets of Boolean functions, J. Soc. Indust. Appl. Math., 11 (1963), 521–525 10.1137/0111037 MR0157826 (28:1055) 0119.01202 LinkISIGoogle Scholar[7] M. A. Harrison, Ph.D. Thesis, Combinatorial problems in Boolean algebras and applications to the theory of switching, University of Michigan, 1963 Google Scholar[8] M. A. Harrison, The number of equivalence classes of Boolean functions under groups containing negation, IEEE Prof. Group of Electronic Computers, (1963), , September 0129.01405 CrossrefGoogle Scholar[9] Michael A. Harrison, The number of classes of invertible Boolean functions, J. Assoc. Comput. Mach., 10 (1963), 25–28 MR0151399 (27:1384) 0126.00901 CrossrefISIGoogle Scholar[10] L. Hellerman, Equivalence classes of logical functions, IBM Tech. Pub. TROO. 819, 1961, November Google Scholar[11] C. S. Lorens, Invertible Boolean functions, Space General Corporation Report, 1962, July Google Scholar[12] K. S. Menger, A modulo two adder for three inputs using a single tunnel diode, I. R. E. Trans., 10 (1961), 530–531 Google Scholar[13] E. I. Nechiporuk, On the synthesis of networks using linear transformations of variables, Dokl. Akad. Nauk. SSSR, 123 (1958), 610–612, Available in English in Automation Express, April 1959, pp. 12–13 0089.12805 Google Scholar[14] G. Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen, und chemische Verbindungen, Acta Math., 68 (1937), 145–253 0017.23202 CrossrefGoogle Scholar[15] G. Pólya, Sur les types des propositions composées, J. Symbolic Logic, 5 (1940), 98–103 MR0002510 (2,65d) 0024.00102 CrossrefGoogle Scholar[16] David Slepian, On the number of symmetry types of Boolean functions of n variables, Canadian J. Math., 5 (1953), 185–193 MR0056560 (15,93f) 0051.24802 CrossrefISIGoogle Scholar[17] David Slepian, Some further theory of group codes, Bell System Tech. J., 39 (1960), 1219–1252 MR0122628 (22:13351) CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Boolean Functions, Invariance Groups, and Parallel Complexity13 July 2006 | SIAM Journal on Computing, Vol. 20, No. 3AbstractPDF (4442 KB) Volume 12, Issue 2| 1964Journal of the Society for Industrial and Applied Mathematics History Submitted:02 October 1962Published online:28 July 2006 InformationCopyright © 1964 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0112026Article page range:pp. 285-299ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

We consider the asymptotic complexity of manipulating matrix groups over finite fields. The question is, given a matrix group G by a list of generators, what can we say in polynomial time about the structure of G? While considerable progress has been made recently in identifying the nonabelian composition factors of a matrix group, the fundamental question of recognizing the simplicity of a non-abelian matrix group remains open. For the purposes of polynomial time computation, we reduce this problem to the following “affine case:” Let A be an elementary abelian p-group which is a non-central minimal normal subgroup of a finite group G. Assume further that the quotient G/A is a finite simple group S of Lie type of characteristic p. The algorithmic goal is to distinguish the simple group S from the nonsimple group G. Both S and G are given as “black-box groups of characteristic p.” The reduction to this basic problem involves a large number of recent techniques and results which we review along the way. We address the affine case for the groups S = PSL(2, q) where q = p , p prime. We show that PSL(2, p) can be recognized in Monte Carlo polynomial time among all black-box groups of known characteristic. The situation for f ≥ 2 seems much harder; we exhibit challenging open cases for every f ≥ 2 when q is large. Along with S = PSL(2, p), the positive result holds for all simple groups S of Lie type of characteristic p with the property that in every nontrivial S-module in characteristic p, every element of S has a fixed point. We call these groups “unisingular.” Very recently, Guralnick, Saxl, and Tiep classified these groups. They found that a large number of classes of simple groups are unisingular, including certain classes of linear and unitary groups, all symplectic groups over an odd prime field, all orthogonal groups of odd degree over an odd prime field, many orthogonal groups of even degree, and a number of classes of exceptional groups. 2 Laszlo Babai and Aner Shalev

Enumeration of orbits on cycles for linear and affine groups

The Erdős-Ko-Rado theorem for the derangement graph of the projective general linear group acting on the projective space

An affine algebraic group

Finite Affine Groups: Cycle Indices, Hall–Littlewood Polynomials, and Probabilistic Algorithms

If G is a permutation group acting on a set Ω, a subset Λ of Ω is called a regular set for G if the set-stabilizer of Λ in G is the identity subgroup. We show here that the projective and affine semi-linear groups acting in the natural way as permutation groups on their respective finite geometries, have, in general, for all finite dimensions and all finite fields, regular sets of points. The exceptions to this are found, and an extension of the results to infinite fields is discussed.

Regular graphs and discrete subgroups of projective linear groups

The spectrum of a finite group is the set of its element orders. An arithmetic criterion determining whether a given natural number belongs to a spectrum of a given group is furnished for all finite special, projective general, and projective special linear and unitary groups.

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type VII. Semisimple classes in PSLn(q) and PSp2n(q)

We call a closed subgroup $G \subseteq \mathop {\mathrm {GL}}\nolimits _{n}({\mathbb{K}})$ a linear Lie group. In this section, we shall use the exponential function to assign to each linear Lie group G a vector space $$\mathop {\bf L{}}\nolimits (G) := \bigl\{ x \in M_n({\mathbb{K}}) \colon \exp({\mathbb{R}}x) \subseteq G\bigr\},$$ called the Lie algebra of G. This subspace carries an additional algebraic structure because, for $x,y \in \mathop {\bf L{}}\nolimits (G)$ , the commutator [x,y]=xy−yx is contained in $\mathop {\bf L{}}\nolimits (G)$ , so that [⋅,⋅] defines a skew-symmetric bilinear operation on $\mathop {\bf L{}}\nolimits (G)$ . As a first step, we shall see how to calculate $\mathop {\bf L{}}\nolimits (G)$ for concrete groups and to use it to generalize the polar decomposition to a large class of linear Lie groups.