scholarly journals On alternating and symmetric groups which are quasi OD-characterizable

2017 ◽  
Vol 16 (04) ◽  
pp. 1750065 ◽  
Author(s):  
Ali Reza Moghaddamfar
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Graph ◽  
Symmetric Groups ◽  
Degree Pattern ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Let [Formula: see text] be the prime graph associated with a finite group [Formula: see text] and [Formula: see text] be the degree pattern of [Formula: see text]. A finite group [Formula: see text] is said to be [Formula: see text]-fold [Formula: see text]-characterizable if there exist exactly [Formula: see text] nonisomorphic groups [Formula: see text] such that [Formula: see text] and [Formula: see text]. The purpose of this paper is two-fold. First, it shows that the symmetric group [Formula: see text] is [Formula: see text]-fold [Formula: see text]-charaterizable. Second, it shows that there exist many infinite families of alternating and symmetric groups, [Formula: see text] and [Formula: see text], which are [Formula: see text]-fold [Formula: see text]-characterizable with [Formula: see text].

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.

OD-Characterization of the Symmetric Group S49

2012 ◽  
Vol 535-537 ◽  
pp. 2596-2599 ◽  
Author(s):  
Yan Xiong Yan
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Prime Graph ◽  
Degree Pattern ◽  
A Finite Group

The degree pattern of a finite group G associated with its prime graph has been introduced in [1] and denoted by D(G). The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |G|=|H| and D(G)=D(H).Moreover, a 1-fold OD-characterizable group is simply called an OD-characterizable group. In this paper, we will show that the symmetric group S49 can be characterized by its order and degree pattern. In fact, the symmetric group S49 is 3-fold OD-characterizable

The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups

2017 ◽  
Vol 16 (02) ◽  
pp. 1750025 ◽  
Author(s):  
Jinke Hai ◽  
Shengbo Ge ◽  
Weiping He
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Nilpotent Group ◽  
Symmetric Groups ◽  
Nilpotent Groups ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
A Finite Group ◽  
The Normalizer Property

Let [Formula: see text] be a finite group and let [Formula: see text] be the holomorph of [Formula: see text]. If [Formula: see text] is a finite nilpotent group or a symmetric group [Formula: see text] of degree [Formula: see text], then the normalizer property holds for [Formula: see text].

A Geometrical Study of the Alternating and Symmetric Groups

1929 ◽  
Vol 25 (2) ◽  
pp. 168-174 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Finite Group ◽  
Hermitian Form ◽  
Symmetric Groups ◽  
Irreducible Group ◽  
Image Position ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

Let a finite group Τ be represented as an irreducible group of order N of linear substitutions on n variables,The variables may be chosen so that the substitutions of the group leave invariant the Hermitian form

scholarly journals Gelfand Pairs Involving the Wreath Product of Finite Abelian Groups with Symmetric Groups

2020 ◽  
pp. 1-7
Author(s):  
Omar Tout
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Abelian Groups ◽  
Symmetric Groups ◽  
Gelfand Pair ◽  
Gelfand Pairs ◽  
Finite Abelian Groups ◽  
A Finite Group

Abstract It is well known that the pair $(\mathcal {S}_n,\mathcal {S}_{n-1})$ is a Gelfand pair where $\mathcal {S}_n$ is the symmetric group on n elements. In this paper, we prove that if G is a finite group then $(G\wr \mathcal {S}_n, G\wr \mathcal {S}_{n-1}),$ where $G\wr \mathcal {S}_n$ is the wreath product of G by $\mathcal {S}_n,$ is a Gelfand pair if and only if G is abelian.

More on the OD-Characterizability of a Finite Group

2011 ◽  
Vol 18 (04) ◽  
pp. 663-674 ◽  
Author(s):  
A. R. Moghaddamfar ◽  
S. Rahbariyan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Alternating Groups ◽  
Degree Pattern ◽  
Prime Graphs ◽  
A Finite Group ◽  
P 53 ◽  
Number Of Authors

The degree pattern of a finite group G was introduced in [10]. We say that G is k-fold OD-characterizable if there exist exactly k non-isomorphic finite groups with the same order and same degree pattern as G. When a group G is 1-fold OD-characterizable, we simply call it OD-characterizable. In recent years, a number of authors attempt to characterize finite groups by their order and degree pattern. In this article, we first show that for the primes p=53, 61, 67, 73, 79, 83, 89, 97, the alternating groups Ap+3 are OD-characterizable, while the symmetric groups Sp+3 are 3-fold OD-characterizable. Next, we show that the automorphism groups Aut (O7(3)) and Aut (S6(3)) are 6-fold OD-characterizable. It is worth mentioning that the prime graphs associated with all these groups are connected.

scholarly journals The invariably generating graph of the alternating and symmetric groups

2020 ◽  
Vol 23 (6) ◽  
pp. 1081-1102
Author(s):  
Daniele Garzoni
Keyword(s):  
Finite Group ◽  
Undirected Graph ◽  
Symmetric Groups ◽  
Conjugacy Classes ◽  
Generating Graph ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

AbstractGiven a finite group G, the invariably generating graph of G is defined as the undirected graph in which the vertices are the nontrivial conjugacy classes of G, and two classes are adjacent if and only if they invariably generate G. In this paper, we study this object for alternating and symmetric groups. The main result of the paper states that if we remove the isolated vertices from the graph, the resulting graph is connected and has diameter at most 6.

OD-Characterization of Symmetric Group S57

2013 ◽  
Vol 834-836 ◽  
pp. 1799-1802
Author(s):  
Mei Yang
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Simple Groups ◽  
Almost Simple Groups ◽  
Degree Pattern ◽  
A Finite Group

In this paper, we show that the symmetric group can be characterized by its order and degree pattern. In fact, we get the following theorem: Let G be a finite group such that and . Then G is isomorphisic to one of the almost simple groups: and . Particularly, is 3-fold OD-characterizable.

The Probabilistic Zeta Function of Alternating and Symmetric Groups

2009 ◽  
Vol 16 (02) ◽  
pp. 195-210
Author(s):  
Andrea Lucchini ◽  
Marilena Massa
Keyword(s):  
Finite Group ◽  
Zeta Function ◽  
Dirichlet Series ◽  
Symmetric Groups ◽  
Simple Zero ◽  
Factor Group ◽  
Explicit Criterion ◽  
Alternating And Symmetric Groups ◽  
A Finite Group

The probability that a finite group G is generated by s elements is given by a truncated Dirichlet series in s, denoted by PG(s). We give an explicit criterion that allows one to recognize whether the factor group G/ Frat (G) is simple by only looking at the coefficients of PG(s). In order to get such a criterion, we prove that the series derived from PG(s) by removing the even-indexed terms has only a simple zero at s=1.

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