Finite Groups Which Factor as Product of an Alternating Group and a Symmetric Group

2004 ◽  
Vol 32 (2) ◽  
pp. 637-647 ◽  
Author(s):  
M. R. Darafsheh
Keyword(s):  
Symmetric Group ◽  
Finite Groups ◽  
Alternating Group

Integral Cayley Graphs over Finite Groups

2020 ◽  
Vol 27 (01) ◽  
pp. 131-136
Author(s):  
Elena V. Konstantinova ◽  
Daria Lytkina
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Cyclic Group ◽  
Finite Groups ◽  
Cayley Graphs ◽  
Alternating Group ◽  
Generating Set ◽  
A Finite Group

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}.

scholarly journals Factorization of groups involving symmetric and alternating groups

2001 ◽  
Vol 27 (3) ◽  
pp. 161-167 ◽  
Author(s):  
M. R. Darafsheh ◽  
G. R. Rezaeezadeh
Keyword(s):  
Symmetric Group ◽  
Finite Groups ◽  
Alternating Group ◽  
Alternating Groups

We obtain the structure of finite groups of the formG=ABwhereBis a group isomorphic to the symmetric group onnlettersSn,n≥5andAis a group isomorphic to the alternating group on 6 letters.

OD-Characterization of Certain Finite Groups Having Connected Prime Graphs

2010 ◽  
Vol 17 (01) ◽  
pp. 121-130 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
A. R. Zokayi
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Finite Groups ◽  
Alternating Group ◽  
Degree Pattern ◽  
Prime Graphs ◽  
Group A ◽  
A Finite Group

The degree pattern of a finite group G denoted by D(G) was introduced in [5]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups having the same order and same degree pattern as G. In the present article, we show that the alternating group A10 and the automorphism group Aut (McL) are 2-fold OD-characterizable, while the automorphism group Aut (J2) is 3-fold OD-characterizable and the symmetric group S10 is 8-fold OD-characterizable. It is worth mentioning that the prime graphs associated to these groups are connected and, in fact, among the groups with this property, they are the first groups which are investigated for OD-characterizability.

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.

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.

COUNTING ELEMENTS IN THE SYMMETRIC GROUP

2009 ◽  
Vol 19 (03) ◽  
pp. 305-313 ◽  
Author(s):  
DAVID EL-CHAI BEN-EZRA
Keyword(s):  
Symmetric Group ◽  
Finite Groups ◽  
Generating Functions ◽  
Symmetric Groups ◽  
Combinatorial Identities ◽  
Subgroup Growth

By using simple ideas from subgroup growth of pro-finite groups we deduce some combinatorial identities on generating functions counting various elements in symmetric groups.

Size of free groups in varieties generated by finite groups

2019 ◽  
Vol 29 (08) ◽  
pp. 1419-1430
Author(s):  
William Cocke
Keyword(s):  
Finite Group ◽  
Free Group ◽  
Finite Groups ◽  
Prime Order ◽  
Free Groups ◽  
Affine Group ◽  
Alternating Group ◽  
A Finite Group

The number of distinct [Formula: see text]-variable word maps on a finite group [Formula: see text] is the order of the rank [Formula: see text] free group in the variety generated by [Formula: see text]. For a group [Formula: see text], the number of word maps on just two variables can be quite large. We improve upon previous bounds for the number of word maps over a finite group [Formula: see text]. Moreover, we show that our bound is sharp for the number of 2-variable word maps over the affine group over fields of prime order and over the alternating group on five symbols.

Galois groups of Taylor polynomials of some elementary functions

2019 ◽  
Vol 15 (06) ◽  
pp. 1127-1141
Author(s):  
Khosro Monsef Shokri ◽  
Jafar Shaffaf ◽  
Reza Taleb
Keyword(s):  
Symmetric Group ◽  
Galois Groups ◽  
Alternating Group ◽  
Integer Number ◽  
Elementary Functions ◽  
Type Formula ◽  
Number Formula ◽  
Taylor Polynomials ◽  
Index 2

Motivated by Schur’s result on computing the Galois groups of the exponential Taylor polynomials, this paper aims to compute the Galois groups of the Taylor polynomials of the elementary functions [Formula: see text] and [Formula: see text]. We first show that the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] are as large as possible, namely, [Formula: see text] (full symmetric group) or [Formula: see text] (alternating group), depending on the residue of the integer number [Formula: see text] modulo [Formula: see text]. We then compute the Galois groups of the [Formula: see text]th Taylor polynomials of [Formula: see text] and show that these Galois groups essentially coincide with the Coexter groups of type [Formula: see text] (or an index 2 subgroup of the corresponding Coexter group).

scholarly journals ALTERNATING AND SYMMETRIC GROUPS WITH EULERIAN GENERATING GRAPH

2017 ◽  
Vol 5 ◽  
Author(s):  
ANDREA LUCCHINI ◽  
CLAUDE MARION
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Number ◽  
Symmetric Groups ◽  
Alternating Group ◽  
Generating Graph ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Given a finite group $G$, the generating graph $\unicode[STIX]{x1D6E4}(G)$ of $G$ has as vertices the (nontrivial) elements of $G$ and two vertices are adjacent if and only if they are distinct and generate $G$ as group elements. In this paper we investigate properties about the degrees of the vertices of $\unicode[STIX]{x1D6E4}(G)$ when $G$ is an alternating group or a symmetric group of degree $n$. In particular, we determine the vertices of $\unicode[STIX]{x1D6E4}(G)$ having even degree and show that $\unicode[STIX]{x1D6E4}(G)$ is Eulerian if and only if $n\geqslant 3$ and $n$ and $n-1$ are not equal to a prime number congruent to 3 modulo 4.

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