Recent zbMATH articles in MSC 20Bhttps://zbmath.org/atom/cc/20B2021-04-16T16:22:00+00:00WerkzeugTwo-distance-primitive graphs.https://zbmath.org/1456.050452021-04-16T16:22:00+00:00"Jin, Wei"https://zbmath.org/authors/?q=ai:jin.wei|jin.wei.1"Wu, Ci Xuan"https://zbmath.org/authors/?q=ai:wu.cixuan"Zhou, Jin Xin"https://zbmath.org/authors/?q=ai:zhou.jinxinSummary: A 2-distance-primitive graph is a vertex-transitive graph whose vertex stabilizer is primitive on both the first step and the second step neighborhoods. Let \(\Gamma\) be such a graph. This paper shows that either \(\Gamma\) is a cyclic graph, or \(\Gamma\) is a complete bipartite graph, or \(\Gamma\) has girth at most \(4\) and the vertex stabilizer acts faithfully on both the first step and the second step neighborhoods. Also a complete classification is given of such graphs satisfying that the vertex stabilizer acts \(2\)-transitively on the second step neighborhood. Finally, we determine the unique 2-distance-primitive graph which is locally cyclic.Automorphisms and isomorphisms of some \(p\)-ary bent functions.https://zbmath.org/1456.941442021-04-16T16:22:00+00:00"Dempwolff, Ulrich"https://zbmath.org/authors/?q=ai:dempwolff.ulrichIn this continuation of the author's paper [Commun. Algebra 34, No. 3, 1077--1131 (2006; Zbl 1085.05019)]
on Boolean (bent) functions, \(p\)-ary bent functions are similarly investigated. EA-equivalence of (bent) functions is in general not easy to decide. Simple invariants, like algebraic degree, are usually not sufficient to decide equivalence of bent functions, stronger methods seem necessary.
In this paper, group-theoretic methods are applied to analyse equivalence of (some classes of) \(p\)-ary bent functions. For a given (bent) function \(f\) from an \(n\)-dimensional vector space \(V\) over \(\mathbb{F}_p\) to \(\mathbb{F}_p\), the author considers the group \(\mathbf{EA}(f)\) of EA automorphisms, i.e. \(\phi_{11} \in (V)\),
\(\phi_{22} \in \mathrm{GL}(\mathbb{F}_p)\), \(\phi_{12} \in \Hom(V,\mathbb{F}_p)\) and \(v\in V, w\in \mathbb{F}_p\) such that
\[ f(\phi_{11}(x)+v) = \phi_{22}(f(x)) + \phi_{12}(x) + w \quad\text{for all }x\in V. \]
The structure of this group is invariant under EA-equivalence.
As another invariant under EA-equivalence for \(p\)-ary functions, the author suggests the set \(\{v\in V, D_v^2f = 0\}\), where \(D_vf(x) = f(x+v)-f(x)\) denotes the derivative of \(f\) in direction \(v\).
In the first part, the author discusses a secondary construction of three types of (non-quadratic) bent functions \(f\), describes \(\mathbf{EA}(f)\) and as a consequence solves the equivalence problem for these types of bent functions.
In the second part of the paper, \(\mathbf{EA}(f)\) is described for Maiorana-McFarland bent functions of the form \(\mathrm{Tr}_n(xy^l)\), \(\gcd(l,p^n-1) = 1\), and the question when two such bent functions are EA-equivalent is solved.
Reviewer: Wilfried Meidl (Linz)Enchanting mazes and commutative groups of permutations.https://zbmath.org/1456.000022021-04-16T16:22:00+00:00"Behrends, Ehrhard"https://zbmath.org/authors/?q=ai:behrends.ehrhard(no abstract)Large orbit sizes in finite group actions.https://zbmath.org/1456.200062021-04-16T16:22:00+00:00"Qian, Guohua"https://zbmath.org/authors/?q=ai:qian.guohua"Yang, Yong"https://zbmath.org/authors/?q=ai:yang.yongLet \(G\) be a finite group acting faithfully on a finite vector space \(V\). The \(G\)-orbit of an element \(v \in V\) is the set \(v^{G}=\{v^{g} \mid g \in G \}\) and the orbit size of \(v\) is \(| v^{G}|\). Results on orbit sizes, particularly on the existence of large orbits, have been fundamental to solving problems in several areas of finite group theory.
In the paper under review, the authors study relations of the sizes of various sections of finite linear groups and the largest orbit size of linear group actions. They also provide various applications of the results they have obtained.
Reviewer: Enrico Jabara (Venezia)Rigid local systems on \(\mathbb{A}^{1}\) with finite monodromy.https://zbmath.org/1456.112322021-04-16T16:22:00+00:00"Katz, Nicholas M."https://zbmath.org/authors/?q=ai:katz.nicholas-mSummary: We formulate some conjectures about the precise determination of the monodromy groups of certain rigid local systems on \(\mathbb{A}^{1}\) whose monodromy groups are known, by results of Kubert, to be finite. We prove some of them.Homogenous finitary symmetric groups.https://zbmath.org/1456.200382021-04-16T16:22:00+00:00"Kegel, Otto H."https://zbmath.org/authors/?q=ai:kegel.otto-h"Kuzucuoğlu, Mahmut"https://zbmath.org/authors/?q=ai:kuzucuoglu.mahmutSummary: We characterize strictly diagonal type of embeddings of finitary symmetric groups in terms of cardinality and the characteristic. Namely, we prove the following. Let \(\kappa\) be an infinite cardinal. If \(G=\bigcup\limits_{i=1}^\infty G_i\), where \(G_i\cong \mathrm{FSym} (\kappa n_i)\), (\(H=\bigcup\limits_{i=1}^\infty H_i\), where \(H_i\cong \mathrm{Alt} (\kappa n_i)\)), is a group of strictly diagonal type and \(\xi=(p_1, p_2, \dots)\) is an infinite sequence of primes, then \(G\) is isomorphic to the homogenous finitary symmetric group \(\mathrm{FSym} (\kappa)(\xi) (H\) is isomorphic to the homogenous alternating group \(\mathrm{Alt} (\kappa)(\xi))\), where \(n_0=1\), \(n_i=p_1p_2\cdots p_i\).Flag-transitive point-primitive symmetric \((v,k,\lambda)\) designs with bounded \(k\).https://zbmath.org/1456.050232021-04-16T16:22:00+00:00"Zhang, Zhilin"https://zbmath.org/authors/?q=ai:zhang.zhilin"Yuan, Pingzhi"https://zbmath.org/authors/?q=ai:yuan.pingzhi"Zhou, Shenglin"https://zbmath.org/authors/?q=ai:zhou.shenglinAuthors' abstract: In [J. Comb. Des. 21, No. 3--4, 127--141 (2013; Zbl 1273.05028)] \textit{D. Tian} and \textit{S. Zhou} conjectured that a flag-transitive and point-primitive automorphism group of a symmetric \((v,k,\lambda)\) design must be an affine or almostsimple group. In this paper, we study this conjecture and prove that if \(k \le 10^3\) and \(G \le \Aut({\mathcal D})\) is flag-transitive and point-primitive, then \(G\) is affine or almost simple.This supports the conjecture.
Reviewer: Dean Crnković (Rijeka)Induced designs and fixed points.https://zbmath.org/1456.050212021-04-16T16:22:00+00:00"Chen, Jianfu"https://zbmath.org/authors/?q=ai:chen.jianfu"Zhou, Shenglin"https://zbmath.org/authors/?q=ai:zhou.shenglinSummary: Let \(\mathcal{D}\) be a 2-\(( v , k , 1 )\) design and \(H\) be an automorphism group of \(\mathcal{D} \). This paper studies the induced design on the fixed points of \(H\), denoted by \(\operatorname{Fix} H\). We extend the work of \textit{A. Camina} and \textit{J. Siemons} [J. Comb. Theory, Ser. A 51, No. 2, 268--276 (1989; Zbl 0675.05008)] and we also state that if there is an induced design on \(\operatorname{Fix} H\), then \(| \operatorname{Fix} H | \leq r\) in most cases, where \(r\) is the number of blocks containing a given point. Moreover, for a given \(k_0\), there exist only finitely many designs admitting an induced design on \(\operatorname{Fix} H\) with \(| \operatorname{Fix} H | > r\) and block size \(k_0\).Generalized Gardiner-Praeger graphs and their symmetries.https://zbmath.org/1456.050782021-04-16T16:22:00+00:00"Miklavič, Štefko"https://zbmath.org/authors/?q=ai:miklavic.stefko"Šparl, Primož"https://zbmath.org/authors/?q=ai:sparl.primoz"Wilson, Stephen E."https://zbmath.org/authors/?q=ai:wilson.stephen-eSummary: A subgroup of the automorphism group of a graph acts half-arc-transitively on the graph if it acts transitively on the vertex-set and on the edge-set of the graph but not on the arc-set of the graph. If the full automorphism group of the graph acts half-arc-transitively, the graph is said to be half-arc-transitive.
\textit{A. Gardiner} and \textit{C. E. Praeger} [Eur. J. Comb. 15, No. 4, 375--381 (1994; Zbl 0806.05037)] introduced two families of tetravalent arc-transitive graphs, called the \(C^{\pm 1}\) and the \(C^{\pm \varepsilon}\) graphs, that play a prominent role in the characterization of the tetravalent graphs admitting an arc-transitive group of automorphisms with a normal elementary abelian subgroup such that the corresponding quotient graph is a cycle. All of the Gardiner-Praeger graphs are arc-transitive but admit a half-arc-transitive group of automorphisms. Quite recently, \textit{P. Potočnik} and \textit{S. E. Wilson} [Art Discrete Appl. Math. 3, No. 1, Paper No. P1.08, 33 p. (2020; Zbl 1441.05109)] introduced the family of \(\operatorname{CPM}\) graphs, which are generalizations of the Gardiner-Praeger graphs. Most of these graphs are arc-transitive, but some of them are half-arc-transitive. In fact, at least up to order 1000, each tetravalent half-arc-transitive loosely-attached graph of odd radius having vertex-stabilizers of order greater than 2 is isomorphic to a \(\operatorname{CPM}\) graph.
In this paper we determine the automorphism group of the \(\operatorname{CPM}\) graphs and investigate isomorphisms between them. Moreover, we determine which of these graphs are 2-arc-transitive, which are arc-transitive but not 2-arc-transitive, and which are half-arc-transitive.The hierarchy of Rameau groups.https://zbmath.org/1456.000772021-04-16T16:22:00+00:00"Jedrzejewski, Franck"https://zbmath.org/authors/?q=ai:jedrzejewski.franckSummary: This paper contributes to the transformational study of progressions of seventh chords and generalizations thereof. PLR transformations are contextual transformations that originally apply only to consonant triads. These transformations were introduced by David Lewin and were inspired the works of musicologist Hugo Riemann. As an alternative to other attempts to define transformations on seventh chords, we define new groups in this article, called Rameau groups, which transform all types of seventh or ninth chords or more generally, any chords formed of stacks of major or minor thirds. These groups form a hierarchy for inclusion. We study on musical examples the ability of these operators to show symmetries in the progression of seventh chords.
For the entire collection see [Zbl 1425.00082].Non-contextual JQZ transformations.https://zbmath.org/1456.000762021-04-16T16:22:00+00:00"Jedrzejewski, Franck"https://zbmath.org/authors/?q=ai:jedrzejewski.franckSummary: Initiated by David Lewin, the contextual PLR-transformations are well known from neo-Riemannian theory. As it has been noted, these transformations are only used for major and minor triads. In this paper, we introduce non-contextual bijections called JQZ transformations that could be used for any kind of chord. These transformations are pointwise, and the JQZ group that they generate acts on any type of n-chord. The properties of these groups are very similar, and the JQZ-group could extend the PLR-group in many situations. Moreover, the hexatonic and octatonic subgroups of JQZ and PLR groups are subdual.
For the entire collection see [Zbl 1425.00082].