scholarly journals A Computational Approach to Verbal Width for Engel Words in Alternating Groups †

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 877
Author(s):  
Jorge Martínez Carracedo
Keyword(s):  
Computational Approach ◽  
Alternating Group ◽  
Arbitrary Length ◽  
Alternating Groups ◽  
Group A ◽  
Engel Word

It is known that every element in the alternating group A n , with n ≥ 5 , can be written as a product of at most two Engel words of arbitrary length. However, it is still unknown if every element in an alternating group is an Engel word of Arbitrary length. In this paper, a different approach to this problem is presented, getting new results for small alternating groups.

Engel words in alternating groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750021 ◽  
Author(s):  
Jorge Martínez Carracedo
Keyword(s):  
Alternating Group ◽  
Arbitrary Length ◽  
Alternating Groups

In this paper, we prove that every element in the alternating group [Formula: see text], with [Formula: see text], can be written as a product of at most two Engel words of arbitrary length.

A CHARACTERIZATION OF ALTERNATING GROUPS BY ORDERS OF SOLVABLE SUBGROUPS

2004 ◽  
Vol 03 (04) ◽  
pp. 445-452 ◽  
Author(s):  
JIANXING BI ◽  
XIANHUA LI
Keyword(s):  
Finite Group ◽  
Alternating Group ◽  
Alternating Groups ◽  
Group A ◽  
A Finite Group

In this paper, we prove that a finite group G is isomorphic to the alternating group An with n≥5 if and only if they have the same set of the orders of solvable subgroups.

ECONOMICAL GENERATING SETS FOR THE SYMMETRIC AND ALTERNATING GROUPS CONSISTING OF CYCLES OF A FIXED LENGTH

2012 ◽  
Vol 11 (06) ◽  
pp. 1250110
Author(s):  
SCOTT ANNIN ◽  
JOSH MAGLIONE
Keyword(s):  
Symmetric Group ◽  
Computer Science ◽  
Theoretical Computer Science ◽  
Alternating Group ◽  
Alternating Groups ◽  
Theoretical Computer ◽  
Fixed Length ◽  
Generating Sets ◽  
Disjoint Cycles ◽  
Group A

The symmetric group Sn and the alternating group An are groups of permutations on the set {0, 1, 2, …, n - 1} whose elements can be represented as products of disjoint cycles (the representation is unique up to the order of the cycles). In this paper, we show that whenever n ≥ k ≥ 2, the collection of all k-cycles generates Sn if k is even, and generates An if k is odd. Furthermore, we algorithmically construct generating sets for these groups of smallest possible size consisting exclusively of k-cycles, thereby strengthening results in [O. Ben-Shimol, The minimal number of cyclic generators of the symmetric and alternating groups, Commun. Algebra35(10) (2007) 3034–3037]. In so doing, our results find importance in the context of theoretical computer science, where efficient generating sets play an important role.

Induced Representations and Alternating Groups

1964 ◽  
Vol 16 ◽  
pp. 587-601 ◽  
Author(s):  
B. M. Puttaswamaiah ◽  
G. de B. ROBINSON
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Alternating Group ◽  
Alternating Groups ◽  
Induced Representations ◽  
University Of Toronto ◽  
Group A ◽  
The University ◽  
Special Case

This paper is based on part of the thesis of one of the authors (5), submitted at the University of Toronto in 1963. In the first part of the paper a result on induced representations (2, 4, 9) is generalized slightly and a number of corollaries are derived. In the rest of the paper a special case of this result is applied to put the representation theory of the alternating group on a par with that of the symmetric group. A knowledge of the representation theory of Sn (7) on the part of the reader is assumed.

Central Idempotents and Units in Rational Group Algebras of Alternating Groups

1998 ◽  
Vol 08 (04) ◽  
pp. 467-477 ◽  
Author(s):  
A. Giambruno ◽  
E. Jespers
Keyword(s):  
Group Algebra ◽  
Finite Index ◽  
Explicit Description ◽  
Alternating Group ◽  
Group Algebras ◽  
Young Tableaux ◽  
Rational Group ◽  
Alternating Groups ◽  
Group Of Units

Let ℚAn be the group algebra of the alternating group over the rationals. By exploiting the theory of Young tableaux, we give an explicit description of the minimal central idempotents of ℚAn. As an application we construct finitely many generators for a subgroup of finite index in the centre of the group of units of ℚAn.

scholarly journals Projective modular representations of finite groups II

1981 ◽  
Vol 22 (1) ◽  
pp. 89-99 ◽  
Author(s):  
J. F. Humphreys
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Theoretical Background ◽  
Alternating Group ◽  
Modular Representations ◽  
Future Paper ◽  
Representations Of Finite Groups ◽  
Group A ◽  
Mathieu Groups

In this paper, which is a continuation of [4], the necessary theoretical background is given to enable the calculation of the irreducible Brauer projective characters of a given finite group to be carried out. As an example, this calculation is done for the alternating group A (7) in §3. In a future paper the calculations for the Mathieu groups will be presented.

scholarly journals The Energy of Conjugacy Classes Graphs of Some Order of Alternating Groups

2020 ◽  
Vol 7 (4) ◽  
pp. 62-71
Author(s):  
Zuzan Naaman Hassan ◽  
Nihad Titan Sarhan
Keyword(s):  
Conjugacy Class ◽  
Adjacency Matrix ◽  
Finite Graph ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Alternating Groups ◽  
Eigen Values ◽  
Finite Set ◽  
Class Graph ◽  
Square Matrix

The energy of a graph , is the sum of all absolute values of the eigen values of the adjacency matrix which is indicated by . An adjacency matrix is a square matrix used to represent of finite graph where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. The group of even permutations of a finite set is known as an alternating group  . The conjugacy class graph is a graph whose vertices are non-central conjugacy classes of a group , where two vertices are connected if their cardinalities are not coprime. In this paper, the conjugacy class of alternating group  of some order for   and their energy are computed. The Maple2019 software and Groups, Algorithms, and Programming (GAP) are assisted for computations.

scholarly journals Convergence of random walks on double transitive group generated by its permutational character

2019 ◽  
Keyword(s):  
Group Algebra ◽  
Transitive Group ◽  
Exact Formula ◽  
Alternating Group ◽  
Stationary Points ◽  
Special Cases ◽  
Group A ◽  
Class Probability ◽  
A Finite Group ◽  
Substitution Group

Let $P$ be a probability on a finite group $G$, $U(g)=\frac{1}{|G|}$ the uniform (trivial) probability on the group $G$, $P^{(n)}=P *\ldots*P$ an $n$-fold convolution of $P$. A lot of estimates of the rate of the convergence $P^{(n)}\rightarrow U$ are found in different norms. It is well known conditions under which $P^{(n)}\rightarrow U$ if $n\rightarrow\infty$. Many papers are devoted to estimating the rate of this convergence for different norms. We consider finite groups that have a double transitive representation by substitutions and the probability that naturally arises in this image. This probability on each element of the group is proportional to the number of fixed (or stationary) points of this element, which is considered as a substitution. In other words, this probability is a character of the substitution representation of the group. A probability is called class if it takes the same values on each class of conjugate elements of a group, that is, it is a function of the class. The considered probability is class because any character of a group takes on the same values on conjugate elements. Any probability (and, in general, functions with values in an arbitrary ring) on a group can be associated with an element of the group algebra of this group over this ring. The class probability corresponds to an element of the center of this group algebra; that is why the class probability is also called central. On an abelian group, any probability is class (central). In the paper convergence with respect to the norm $\|F\|=\sum\limits_{g\in G} |F(g)|$, where $F(g)$ is a function on group $G$, is considered. For the norm an exact formula not estimate only, as usual for rate of convergence of convolution $P^{(n)}\rightarrow U$ is given. It turns out that the norm of the difference $\|P^{(n)}-U\|$ is determined by the order of the group, degree the group as a substitution group, and the number of regular substitutions in the group. A substitution is called regular if it has no fixed points. Special cases are considered the symmetric group, the alternating group, the Zassenhaus group, and the Frobenius group of order $p(p-1)$ with the Frobenius core of order $p$ ($p$ is a prime number). A Zassenhaus group is a double transitive substitution group of a finite set in which only a trivial substitution leaves more than two elements of this set fixed.

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