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We consider the so-called Derksen group which is a subgroup of the polynomial automorphism group of the polynomial ring in n variables over a field. The Derksen group is generated by affine automorphisms and one particular non-linear automorphism. Derksen (1994) proved that if the characteristic of the underlying field is zero and n ≥ 3, then the Derksen group is equal to the entire tame subgroup. The result is called Derksen’s Theorem. It is quite natural to ask whether the same property holds for positive characteristic. In this paper, we point out that the question can be easily answered negatively when the underlying field is of characteristic two. We shall also prove that the permutation group induced by the Derksen group over a finite field of characteristic two is a proper subgroup of the alternating group on the n-dimensional linear space over the finite field. This is a stronger result that Derksen’s Theorem does not hold when the underlying field is a finite field of characteristic two.

The Eulerian polynomial $$ \mathrm {AExc}_n(t)$$ enumerating excedances in the symmetric group $$\mathfrak {S}_n$$ is known to be gamma positive for all n. When enumeration is done over the type B and type D Coxeter groups, the type B and type D Eulerian polynomials are also gamma positive for all n. We consider $$ \mathrm {AExc}_n^+(t)$$ and $$ \mathrm {AExc}_n^-(t)$$ , the polynomials which enumerate excedance in the alternating group $$\mathcal {A}_n$$ and in $$\mathfrak {S}_n - \mathcal {A}_n$$ , respectively. We show that $$ \mathrm {AExc}_n^+(t)$$ is gamma positive iff $$n \ge 5$$ is odd. When $$n \ge 4$$ is even, $$ \mathrm {AExc}_n^+(t)$$ is not even palindromic, but we show that it is the sum of two gamma positive summands. An identical statement is true about $$ \mathrm {AExc}_n^-(t)$$ . We extend similar results to the excedance based Eulerian polynomial when enumeration is done over the positive elements in both type B and type D Coxeter groups. Gamma positivity results are known when excedance is enumerated over derangements in $$\mathfrak {S}_n$$ . We extend some of these to the case when enumeration is done over even and odd derangements in $$\mathfrak {S}_n$$ .

In this paper, we propose a recursive algorithm and a parallel algorithm for constructing independent spanning trees in ANn. The main ideas of the recursive algorithm are to construct large trees f...

Highly nonlinear functions (perfect nonlinear, maximum nonlinear, etc.) between finite fields have been studied in numerous papers. These concepts have been generalized first to finite abelian groups and then to finite nonabelian groups. In 2011, Poinsot and Pott introduced a new class of highly nonlinear functions between finite nonabelian groups, which are closely related to maximum nonlinear functions. They found such a function by a computer search, and proposed to find more such functions by theoretical constructions. Since then there is no progress in this topic. In this paper we continue the research in the direction proposed by Poinsot and Pott. Our purpose is to study the properties and constructions of this new class of highly nonlinear functions, called the absolute maximum nonlinear functions in this paper. In particular, for arbitrary finite (abelian or nonabelian) groups $K$ and $N$ with an absolute maximum nonlinear function $f: K \to N$ , we will prove that $N$ has to be abelian, and $f$ cannot be perfect nonlinear if $K$ is not abelian. Then we investigate the numerical constraints on the groups that have absolute maximum nonlinear functions. These constraints will eliminate the existence of absolute maximum nonlinear functions for many finite nonabelian groups. More importantly, using these constraints we will develop a method to construct absolute maximum nonlinear functions for some groups. In particular, we will determine all absolute maximum nonlinear functions on $A_4$ (the alternating group of degree 4). Furthermore, a general method to construct new absolute maximum nonlinear functions from the old ones will be given.

Given a connected graph $G$ and a positive integer $\ell $, the $\ell $-extra (resp. $\ell $-component) edge connectivity of $G$, denoted by $\lambda ^{(\ell )}(G)$ (resp. $\lambda _{\ell }(G)$), is the minimum number of edges whose removal from $G$ results in a disconnected graph so that every component has more than $\ell $ vertices (resp. so that it contains at least $\ell $ components). This naturally generalizes the classical edge connectivity of graphs defined in term of the minimum edge cut. In this paper, we proposed a general approach to derive component (resp. extra) edge connectivity for a connected graph $G$. For a connected graph $G$, let $S$ be a vertex subset of $G$ for $G\in \{\Gamma _{n}(\Delta ),AG_n,S_n^2\}$ such that $|S|=s\leq |V(G)|/2$, $G[S]$ is connected and $|E(S,G-S)|=\min \limits _{U\subseteq V(G)}\{|E(U, G-U)|: |U|=s, G[U]\ \textrm{is connected}\ \}$, then we prove that $\lambda ^{(s-1)}(G)=|E(S,G-S)|$ and $\lambda _{s+1}(G)=|E(S,G-S)|+|E(G[S])|$ for $s=3,4,5$. By exploring the reliability analysis of $AG_n$ and $S_n^2$ based on extra (component) edge faults, we obtain the following results: (i) $\lambda _3(AG_n)-1=\lambda ^{(1)}(AG_n)=4n-10$, $\lambda _4(AG_n)-3=\lambda ^{(2)}(AG_n)=6n-18$ and $\lambda _5(AG_n)-4=\lambda ^{(3)}(AG_n)=8n-24$; (ii) $\lambda _3(S_n^2)-1=\lambda ^{(1)}(S_n^2)=4n-8$, $\lambda _4(S_n^2)-3=\lambda ^{(2)}(S_n^2)=6n-15$ and $\lambda _5(S_n^2)-4=\lambda ^{(3)}(S_n^2)=8n-20$. This general approach maybe applied to many diverse networks.

We introduce the alternating Schur algebra $AS_F(n,d)$ as the commutant of the action of the alternating group $A_d$ on the $d$-fold tensor power of an $n$-dimensional $F$-vector space. When $F$ has characteristic different from $2$, we give a basis of $AS_F(n,d)$ in terms of bipartite graphs, and a graphical interpretation of the structure constants. We introduce the abstract Koszul duality functor on modules for the even part of any $\mathbf Z/2\mathbf Z$-graded algebra. The algebra $AS_F(n,d)$ is $\mathbf Z/2\mathbf Z$-graded, having the classical Schur algebra $S_F(n,d)$ as its even part. This leads to an approach to Koszul duality for $S_F(n,d)$-modules that is amenable to combinatorial methods. We characterize the category of $AS_F(n,d)$-modules in terms of $S_F(n,d)$-modules and their Koszul duals. We use the graphical basis of $AS_F(n,d)$ to study the dependence of the behavior of derived Koszul duality on $n$ and $d$., 27 pages, 1 figure, to appear in Pacific Journal of Mathematics

Combined-permutation representations (CPRs) for characterizing $${\varvec{I}}_{h}$$ -skeletons (a dodecahedral skeleton 1 having 20 vertices, an icosahedral skeleton 2 having 12 vertices, and a $$\hbox {C}_{{60}}$$ -fullerene skeleton 3 having 60 vertices) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to give the CPR of degree 22 (= 20 + 2) for 1, the CPR of degree 14 (= 12 + 2) for 2, and the CPR of degree 62 (= 60 + 2) for 3. Mark tables (tables of marks) of these CPRs are different in the sequence of subgroups from each other when they are produced as primary mark tables by the GAP system. On the other hand, the GAP functions MarkTableforUSCI and constructUSCITable, which have been previously developed to systematize the concordant construction of a standard mark table and a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table, are capable of constructing the standard mark table and the standard USCI-CF table even if we start from any of these CPRs. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (Fujita in Symmetry and combinatorial enumeration in chemistry, Springer, Berlin, 1991). Construction of the CPR of degree 7 (= 5 + 2) for characterizing the $${\varvec{I}}_{h}$$ group is also discussed by starting from the alternating group $$\hbox {A}_{{5}}$$ isomorphic to the point group $${\varvec{I}}$$ .

We essentially determine the saturated 2‐fusion systems of J‐component type in which the centralizer of some fully centralized involution of maximal 2‐rank contains a component that is the 2‐fusion system of an alternating group A_n for some n ⩾ 8.

For a connected graph $$G=(V,E)$$ , two spanning trees $$T_1$$ and $$T_2$$ of G are said to be a pair of completely independent spanning trees (or a dual-CIST for short) if for any two vertices $$u,v\in V$$ , the paths joining u and v in the two trees have no common vertex except for u and v. Although the existence of a dual-CIST in the underlying graph of a network has the practical application of protection routing on fault-tolerance, it has been proved that determining whether a graph G admits a dual-CIST is NP-complete. As we know that Cayley graphs are a large family of graphs, some of its subclasses have been attracted and thus graphs in these subclasses have been adopted as the topologies of interconnection networks, such as the n-dimensional star graphs $$S_n$$ , bubble sort graphs $$BS_n$$ , pancake graph $$P_n$$ , alternating group networks $$AGN_n$$ and so on. Pai and Chang (IEEE/ACM Trans Netw 27(3): 1112–1123, 2019) recently showed that there exist dual-CISTs in $$S_n$$ , $$BS_n$$ , $$AGN_n$$ for $$n\geqslant 5$$ and provided their corresponding protection routings. So far, the problem of constructing dual-CISTs on $$P_n$$ has not been dealt with yet. In this sequel, we continue the investigation of the construction of dual-CISTs in pancake graphs as a complementary result. Since $$P_n$$ , $$S_n$$ , and $$BS_n$$ are with the same scale, we experimentally assess the performance of protection routing through simulation results for comparing them when $$n=5,6,7$$ .

We report a new three-point compact sixth-order approximation for the solution of nonlinear two-point boundary value problems $$ {-}w^{\prime\prime} + f\left( {t,w} \right) = 0 $$, subject to natural boundary conditions prescribed at two end points. We also discuss the application of single-step alternating group explicit (SAGE) iteration method to the nonlinear difference equation as a computational tool. The error analysis of the SAGE iteration method is discussed briefly. We have compared the results obtained by using the proposed SAGE iteration method with the results obtained by corresponding two-step alternating group explicit iteration method to demonstrate experimentally the superiority of the proposed method.

It is proven that, for a wide range of integers s (2 < s < p − 2), the existence of a single wildly ramified odd prime l ≠ p leads to either the alternating group or the full symmetric group as Galois group of any irreducible trinomial Xp + aXs + b of prime degree p.

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

We obtain a formula for the orders of the relatively free groups of finite rank in the variety generated by the alternating group [Formula: see text]. This is accomplished by explicitly determining the structure of the commutator subgroups of these groups.

This thesis concerns combinatorial properties of the modular representation theory of the symmetric and alternating groups.We focus on the problem of labelling the modular irreducible representations of the symmetric and alternating groups. A natural way to approach this is through unitriangular basic sets for the decomposition matrices. One of our main motivations is related to the Mullineux map, which controls the restriction from the symmetric group to the alternating subgroup. In order to have a labelling which is particularly well adapted to this restriction, we look for stable unitriangular basic sets. Such sets are stable for conjugation.In our study, the following remarkable phenomenon is observed : the number of self-conjugate partitions with diagonal hook-lengths not divisible by p, called BG-partitions, is equal to the number of fixed points of the Mullineux map, or self-Mullineux partitions. We give a combinatorial and explicit correspondence between the two sets of partitions. Recently it has been shown that there is not always a unitriangular basic set for the alternating group. However, these notions can be defined at the level of blocks. In this thesis, we study the Mullineux map in self-conjugate blocks of p-weight 2 of the symmetric group and we construct a stable unitriangular basic set for these blocks, which implies the existence of unitriangular basic sets for some blocks of the alternating groups., Dans cette thèse nous nous intéressons aux propriétés combinatoires de la théorie des représentations modulaires des groupes symétriques et alternés.Nous nous concentrons sur le problème de l’étiquetage des représentations irréductibles modulaires des groupes symétriques et alternés. Une façon naturelle d’aborder ce problème est de trouver des ensembles basiques unitriangulaires pour les matrices de décomposition. Une de nos principales motivations est liée à l’application de Mullineux, application qui contrôle la restriction du groupe symétrique au sous-groupe alterné. Dans le but d’avoir un étiquetage particulièrement bien adapté à cette restriction, on cherche des ensembles basiques unitriangulaires stables. De tels ensembles sont stables pour la conjugaison.Il apparaît lors de notre étude le phénomène remarquable suivant : le nombre de partitions auto-conjuguées avec des longueurs des crochets diagonaux non divisibles par p, appelées BG-partitions, est égal au nombre de points fixes de l’application de Mullineux, ou partitions auto-Mullineux. Nous montrons une correspondance combinatoire explicite entre les deux ensembles de partitions.Récemment il a été montré qu’il n’existe pas toujours un ensemble basique unitriangulaire pour le groupe alterné. Cependant, ces notions peuvent être définies au niveau des blocs. Nous étudions l’application de Mullineux dans les blocs auto-conjugués de p-poids 2 du groupe symétrique et nous exhibons un ensemble basique unitriangulaire stable pour ces blocs, ce qui implique l’existence d’ensembles basiques unitriangulaires pour certains blocs des groupes alternés.

We prove that an invariant subalgebra A n W of the Weyl algebra A n is a Galois order over an adequate commutative subalgebra Γ when W is a two-parameters irreducible unitary reflection group G ( m , 1 , n ) , m ≥ 1 , n ≥ 1 , including the Weyl group of type B n , or alternating group A n , or the product of n copies of a cyclic group of fixed finite order. Earlier this was established for the symmetric group in [15] . In each of the cases above, except for the alternating groups, we show that A n W is free as a right (left) Γ-module. Similar results are established for the algebra of W-invariant differential operators on the n-dimensional torus where W is a symmetric group S n or orthogonal group of type B n or D n . As an application of our technique we prove the quantum Gelfand-Kirillov conjecture for U q ( s l 2 ) , the first Witten deformation and the Woronowicz deformation.

International audience; A set of spanning trees in a graph G is called independent spanning trees (ISTs) if they are rooted at the same vertex r, and for each vertex v(= r) in G, the two paths from v to r in any two trees share no common vertex expect for v and r. ISTs can be applied in many research fields, such as fault-tolerant broadcasting and secure message distribution in reliable communication networks. Since Cayley graphs have been widely used to design interconnection networks, constructing ISTs on cayley graphs is worth studying. The alternating group network is a subclass of Cayley graphs, and the approach of constructing ISTs in alternating group networks is still unknown. In this paper, we propose a novel and simple top-down approach for constructing ISTs in alternating group networks. The main ideas of the algorithm are to use induction to develop small trees to big trees, to use a triangle breadth-first search (TBFS) process to create a backbone of an IST, and to use breadth-first search (BFS) process to connect the rest of nodes. Compared to other methods of different interconnection networks in the literature, the uniqueness of our method is that it does not need to determine the parent of one node by any rule, on the contrary others determine that by rules. The time complexity in n-dimensional alternating group network AN n is O(n 2 × n!), where n! is twice the number of nodes of AN n ; hence it is polynomial time. We implement the algorithm in PHP and run cases from AN 3 to AN 10. The results reveal that all spanning trees of AN 3 to AN 10 are independent and that our algorithm is accurate and efficient. INDEX TERMS Independent spanning trees, alternating group networks, triangle breadth-first search, interconnection networks, Cayley graph.

The size of a minimal generating set for the commutator subgroup of Sylow 2-subgroups of alternating group is found. The structure of commutator subgroup of Sylow 2-subgroups of the alternating group ${A_{{2^{k}}}}$ is investigated. It is shown that $(Syl_2 A_{2^k})^2 = Syl'_2 A_{2^k}, \, k>2$. It is proved that the commutator length of an arbitrary element of the iterated wreath product of cyclic groups $C_{p_i}, \, p_i\in \mathbb{N}$ equals to 1. The commutator width of direct limit of wreath product of cyclic groups is found. This paper presents upper bounds of the commutator width $(cw(G))$ of a wreath product of groups. A recursive presentation of Sylows $2$-subgroups $Syl_2(A_{{2^{k}}})$ of $A_{{2^{k}}}$ is introduced. As a result the short proof that the commutator width of Sylow 2-subgroups of alternating group ${A_{{2^{k}}}}$, permutation group ${S_{{2^{k}}}}$ and Sylow $p$-subgroups of $Syl_2 A_{p^k}$ ($Syl_2 S_{p^k}$) are equal to 1 is obtained. A commutator width of permutational wreath product $B \wr C_n$ is investigated. An upper bound of the commutator width of permutational wreath product $B \wr C_n$ for an arbitrary group $B$ is found.

Suppose that G is a finite group, π(G) is the set of prime divisors of its order, and ω(G) is the set of orders of its elements. A graph with the following adjacency relation is defined on π(G): different vertices r and s from π(G) are adjacent if and only if rs ∈ ω(G). This graph is called the Gruenberg—Kegel graph or the prime graph of G and is denoted by GK(G). In A. V. Vasil’ev’s Question 16.26 from the “Kourovka Notebook,” it is required to describe all pairs of nonisomorphic simple nonabelian groups with identical Gruenberg—Kegel graphs. M. Hagie and M. A. Zvezdina gave such a description in the case where one of the groups coincides with a sporadic group and an alternating group, respectively. The author solved this question for finite simple groups of Lie type over fields of the same characteristic. In the present paper, we prove the following theorem. Theorem. Let$$G = A_{n - 1}^ \pm \left( q \right)$$, where n ∈{3, 4, 5, 6}, and let G1be a finite simple group of Lie type over a field of order q1nonisomorphic to G, where q = pf, $${q_1} = p_1^{{f_1}}$$, and p and p1are different primes. If the graphs GK(G) and GK(G1) coincide, then one of the following statements holds (1) {G, G1} = {A1(7), A1(8)} (2) {G, G1} = {A3(3), 2F4(2)′} (3) {G, G1} = {2A3(3), A1(49)} (4) {G, G1} = {A2(q), 3D4(q1)}, where (q − 1)3 ≠ 3, q + 1 ≠ 2k, and q1 > 2 (5) $$\left\{G, G_{1}\right\}=\left\{A_{4}^{\varepsilon}(q), A_{4}^{\varepsilon_{1}}\left(q_{1}\right)\right\}$$, where qq1is odd (6) $$\left\{ {G,{G_1}} \right\} = \left\{ {A_4^\varepsilon (q){,^3}{D_4}\left( {{q_1}} \right)} \right\}$$, where (q − ϵ1)5 ≠ 5 and q1 > 2 (7) $${A_4^\varepsilon (q)}$$ and G1 ∈ {B3(q1), C3(q1), D4(q1)}.

The paper is concerned with systems of generators of permutation groups on Cartesian products of residue rings. Each separate permutation from the system of generators is constructed on the basis of additions, is characterized by the local action, and leaves fixed the major parts of the components of the element being transformed. A criterion of 2-transitivity of the generated permutation group is given in the form of the strong connectedness of the digraph which corresponds to the system of generators and which is defined on the set of numbers of residue rings in the Cartesian product. Necessary and sufficient conditions under which this group contains an alternating group are formulated.

We show that a rank reduction technique for string C-group representations first used in [Adv. Math. 228 (2018), pp. 3207-3222] for the symmetric groups generalizes to arbitrary settings. The technique permits us, among other things, to prove that orthogonal groups defined on d-dimensional modules over fields of even order greater than 2 possess string C-group representations of all ranks 3 ≤ n ≤ d. The broad applicability of the rank reduction technique provides fresh impetus to construct, for suitable families of groups, string C-groups of highest possible rank. It also suggests that the alternating group Alt(11)-the only known group having "rank gaps"-is perhaps more unusual than previously thought., SCOPUS: ar.j, info:eu-repo/semantics/published

We use character theory and character estimates to show that the number of T 2 {T_{2}} -systems of the alternating group A n {A_{n}} is at least 1 8 ⁢ n ⁢ 3 ⁢ exp ⁡ ( 2 ⁢ π 6 ⁢ n 1 / 2 ) ⁢ ( 1 + o ⁢ ( 1 ) ) . \frac{1}{8n\sqrt{3}}\exp\biggl{(}\frac{2\pi}{\sqrt{6}}n^{1/2}\biggr{)}(1+o(1)). Applying this result, we obtain a lower bound for the number of connected components of the product replacement graph Γ 2 ⁢ ( A n ) {\Gamma_{2}(A_{n})} .

In this paper, based on the calculation using GAP, we give a classification result on arc-transitive Cayley digraphs of finite simple groups. Let G be a finite simple group and S ⊂ G ∖ { 1 } with | S | = 3 , S ≠ S − 1 and G = 〈 S S − 1 〉 . If the Cayley digraph Γ = Cay ( G , S ) is arc-transitive, then Γ is either normal or isomorphic to one of 382 Cayley digraphs of the alternating group A 47 . Further, we consider the underlying graphs and standard double covers of these 382 Cayley digraphs, and then we get 172 (non-isomorphic) half-transitive Cayley graphs of valency 6, and 144 semisymmetric cubic graphs.

A nite group G is said to be (l; m; n)-generated, if it is a quotient group of the triangle group T(l; m; n) = ⟨ x; y; zlxl = ym = zn = xyz = 1⟩ : In [28], Moori posed the question of nding all the (p; q; r) triples, where p; q and r are prime numbers, such that a non-abelian nite simple group G is (p; q; r)- generated. In this paper we will establish all the (p; q; r)-generations of the alternating group A10: GAP [24] and the Atlas of nite group representations [1] are used in our computations.Key words: Conjugacy classes, generation, structure constant, alternating groups.

This study tackles the relationship between the alternating group 14An"> and young’sdiagrams concerning standard rows. The study has been divided into two stages… First, even permutations have been found depending on the conception of partition 14خ¼"> through formulating an algorithm for this purpose. Secondly, the relationship between the cycle length and partition 14خ¼"> has been found. 14د¨ .1"> Introduction: Let 14n"> be a non-negative integer, a composition 14 خ¼"> of 14n"> is a sequence 14 خ¼=خ¼1 ,خ¼2, ⋯خ¼r "> of non-negative integers such that 14 خ¼=j=1rخ¼j =n"> , [5]. For example, if 14n=4"> , the following sequences are compositions: 144, 3,1, 2,2, 1,3,2,1,1,1,2,1,1,1,2,1,1,1,1"> .

It is shown that each subgroup of odd index in an alternating group of degree at least 10 has all insoluble composition factors to be alternating. A classification is then given of 2-arc-transitive graphs of odd order admitting an alternating group or a symmetric group. This is the second of a series of papers aiming towards a classification of 2-arc-transitive graphs of odd order., 12 pages

International audience; The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this work, we give an explicit bijection between the two families of partitions in terms of the Mullineux symbol.