scholarly journals Product of Conjugacy Classes of the Alternating Group An

2012 ◽  
Vol 9 (3) ◽  
pp. 565-568
Author(s):  
Baghdad Science Journal
Keyword(s):  
Positive Integer ◽  
Conjugacy Class ◽  
Symmetric Group ◽  
Nonempty Subset ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Class C

For a nonempty subset X of a group G and a positive integer m , the product of X , denoted by Xm ,is the set Xm = That is , Xm is the subset of G formed by considering all possible ordered products of m elements form X. In the symmetric group Sn, the class Cn (n odd positive integer) split into two conjugacy classes in An denoted Cn+ and Cn- . C+ and C- were used for these two parts of Cn. This work we prove that for some odd n ,the class C of 5- cycle in Sn has the property that = An n 7 and C+ has the property that each element of C+ is conjugate to its inverse, the square of each element of it is the element of C-, these results were used to prove that C+ C- = An exceptional of I (I the identity conjugacy class), when n=5+4k , k>=0.

Conjugacy classes of double covers of monomial groups

1984 ◽  
Vol 96 (2) ◽  
pp. 195-201 ◽  
Author(s):  
John F. Humphreys
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Symmetric Group ◽  
Double Cover ◽  
Conjugacy Classes ◽  
Alternating Group ◽  
Double Covers ◽  
Monomial Group ◽  
The Subject ◽  
A Finite Group

Let G be a finite group, Sn be the symmetric group on n symbols and An be the corresponding alternating group. The conjugacy classes of the wreath product GSn (or monomial group as it is sometimes known) and the conjugacy classes of GAn have been described by Kerber (see [2] and [3]). The group Sn has a double cover n so that the faithful complex representations of this double cover may be regarded as protective representations of Sn. In Section 2, a particular double cover for GSn is constructed, the faithful complex representations of this group being the subject of a joint article with Peter Hoffman[1]. In the present paper, our task is to determine whether a conjugacy class of GSn corresponds to one or to two conjugacy classes in the double cover of GSn (and similarly for GAn). The main results, Theorems 1 and 2, are stated precisely in Section 2 and proved in Sections 3 and 4 respectively. The case when G = 1 provides classical results of Schur ([5], Satz IV). When G is a cyclic group, Read [4] has determined the conjugacy classes, not just for our particular double cover, but for all possible double covers of GSn.

A Class of Trinomials with Galois Group Sn

2012 ◽  
Vol 19 (spec01) ◽  
pp. 905-911 ◽  
Author(s):  
Anuj Bishnoi ◽  
Sudesh K. Khanduja
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Group ◽  
Rational Numbers ◽  
Alternating Group ◽  
Polynomial Formula

A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.

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.

Infinite families of monogenic trinomials and their Galois groups

2018 ◽  
Vol 29 (05) ◽  
pp. 1850039 ◽  
Author(s):  
Lenny Jones ◽  
Tristan Phillips
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Groups ◽  
Alternating Group

Let [Formula: see text] with [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote, respectively, the symmetric group and alternating group on [Formula: see text] letters. Let [Formula: see text] be an indeterminate, and define [Formula: see text] where [Formula: see text] are certain prescribed forms in [Formula: see text]. For a certain set of these forms, we show unconditionally that there exist infinitely many primes [Formula: see text] such that [Formula: see text] is irreducible over [Formula: see text], [Formula: see text], and the fields [Formula: see text] are distinct and monogenic, where [Formula: see text]. Using a different set of forms, we establish a similar result for all square-free values of [Formula: see text], with [Formula: see text], and any positive integer value of [Formula: see text] for which [Formula: see text] is square-free. Additionally, in this case, we prove that [Formula: see text]. Finally, we show that these results can be extended under the assumption of the [Formula: see text]-conjecture. Our methods make use of recent results of Helfgott and Pasten.

Groups in which squares have boundedly many conjugates

2019 ◽  
Vol 22 (1) ◽  
pp. 133-136
Author(s):  
Gláucia Dierings ◽  
Pavel Shumyatsky
Keyword(s):  
Positive Integer ◽  
Conjugacy Class ◽  
Present Article ◽  
Conjugacy Classes ◽  
Famous Result ◽  
Bounded Size

Abstract Given a group G, we write {x^{G}} for the conjugacy class of G containing the element x. A famous result of B. H. Neumann states that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group {G^{\prime}} is finite. Recently we showed that if {|x^{G}|\leq n} for any commutator x, then {|G^{\prime\prime}|} is finite and n-bounded. If {|x^{G^{\prime}}|\leq n} for any commutator x, then {|\gamma_{3}(G^{\prime})|} is finite and n-bounded. The present article deals with groups in which the conjugacy classes containing squares are finite with bounded size. The following theorem is proved. Let n be a positive integer, G a group and H the subgroup generated by all squares in G. If {|x^{H}|\leq n} for any square {x\in G} , then the order of {\gamma_{3}(H)} is finite and n-bounded.

scholarly journals Covering Theorems for FINASIGS VIII—almost all conjugacy classes in An have exponent ≤4

1978 ◽  
Vol 25 (2) ◽  
pp. 210-214 ◽  
Author(s):  
J. L. Brenner
Keyword(s):  
Conjugacy Class ◽  
Conjugacy Classes ◽  
The Other ◽  
Alternating Group ◽  
Other Hand ◽  
Covering Theorems ◽  
Image Position ◽  
Almost All

AbstractThe product of two subsets C, D of a group is defined as . The power Ce is defined inductively by C0 = {1}, Ce = CCe−1 = Ce−1C. It is known that in the alternating group An, n > 4, there is a conjugacy class C such that CC covers An. On the other hand, there is a conjugacy class D such that not only DD≠An, but even De≠An for e<[n/2]. It may be conjectured that as n ← ∞, almost all classes C satisfy C3 = An. In this article, it is shown that as n ← ∞, almost all classes C satisfy C4 = An.

scholarly journals Sign Conjugacy Classes of the Symmetric Groups

10.37236/5060 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Lucia Morotti
Keyword(s):  
Finite Group ◽  
Conjugacy Class ◽  
Irreducible Character ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Class C ◽  
A Finite Group

A conjugacy class $C$ of a finite group $G$ is a sign conjugacy class if every irreducible character of $G$ takes value 0, 1 or -1 on $C$. In this paper we classify the sign conjugacy classes of the symmetric groups and thereby verify a conjecture of Olsson.

ON THE REGULAR GRAPH RELATED TO THE G-CONJUGACY CLASSES

2021 ◽  
pp. 1-5
Author(s):  
SH. RAHIMI ◽  
Z. AKHLAGHI
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Conjugacy Class ◽  
Regular Graph ◽  
Simple Graph ◽  
Conjugacy Classes ◽  
A Finite Group

Abstract Given a finite group G with a normal subgroup N, the simple graph $\Gamma _{\textit {G}}( \textit {N} )$ is a graph whose vertices are of the form $|x^G|$ , where $x\in {N\setminus {Z(G)}}$ and $x^G$ is the G-conjugacy class of N containing the element x. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not coprime. We prove that, if $\Gamma _G(N)$ is a connected incomplete regular graph, then $N= P \times {A}$ where P is a p-group, for some prime p, $A\leq {Z(G)}$ and $\textbf {Z}(N)\not = N\cap \textbf {Z}(G)$ .

THE CONJUGACY CLASSES OF SOME FINITE METABELIAN GROUPS AND THEIR RELATED GRAPHS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Nor Haniza Sarmin ◽  
Ain Asyikin Ibrahim ◽  
Alia Husna Mohd Noor ◽  
Sanaa Mohamed Saleh Omer
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Dihedral Group ◽  
Clique Number ◽  
Conjugacy Classes ◽  
Quaternion Group ◽  
Metabelian Groups ◽  
Dominating Number ◽  
Graph Properties ◽  
Class Graph

In this paper, the conjugacy classes of three metabelian groups, namely the Quasi-dihedral group, Dihedral group and Quaternion group of order 16 are computed. The obtained results are then applied to graph theory, more precisely to conjugate graph and conjugacy class graph. Some graph properties such as chromatic number, clique number, dominating number and independent number are found.   

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