scholarly journals Minimal permutation representations of the two double covers of Sn

1996 ◽  
Vol 39 (2) ◽  
pp. 285-289
Author(s):  
John Brinkman
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Symmetric Group ◽  
Double Cover ◽  
Permutation Representation ◽  
Double Covers ◽  
A Finite Group

Let G be a finite group and denote by µ(G) (see [2]) the least positive integer m such that G has a faithful permutation representation in the symmetric group of degree m. This note considers the value of µ(G) when G is a double cover of the symmetric group.

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.

scholarly journals On minimal faithful permutation representations of finite groups

2000 ◽  
Vol 62 (2) ◽  
pp. 311-317 ◽  
Author(s):  
L. G. Kovács ◽  
Cheryl E. Praeger
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Positive Integer ◽  
Symmetric Group ◽  
Finite Groups ◽  
Abelian Normal Subgroup ◽  
Minimal Counterexample ◽  
Representations Of Finite Groups ◽  
A Finite Group

The minimal faithful permutation degree μ(G) of a finite group G is the least positive integer n such that G is isomorphic to a subgroup of the symmetric group Sn. Let N be a normal subgroup of a finite group G. We prove that μ(G/N) ≤ μ(G) if G/N has no nontrivial Abelian normal subgroup. There is an as yet unproved conjecture that the same conclusion holds if G/N is Abelian. We investigate the structure of a (hypothetical) minimal counterexample to this conjecture.

scholarly journals THE SUBRING OF GROUP COHOMOLOGY CONSTRUCTED BY PERMUTATION REPRESENTATIONS

2002 ◽  
Vol 45 (1) ◽  
pp. 241-253 ◽  
Author(s):  
David J. Green ◽  
Ian J. Leary ◽  
Björn Schuster
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Group Cohomology ◽  
Subject Classification ◽  
Permutation Representation ◽  
Primary 20J06 ◽  
Subgroup Structure ◽  
A Finite Group

AbstractEach permutation representation of a finite group $G$ can be used to pull cohomology classes back from a symmetric group to $G$. We study the ring generated by all classes that arise in this fashion, describing its variety in terms of the subgroup structure of $G$.We also investigate the effect of restricting to special types of permutation representations, such as $\mathrm{GL}_n(\mathbb{F}_p)$ acting on flags of subspaces.AMS 2000 Mathematics subject classification: Primary 20J06

scholarly journals Groups fixing graphas in switching classes

1982 ◽  
Vol 33 (1) ◽  
pp. 76-85
Author(s):  
Martin W. Liebeck
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Permutation Group ◽  
Abelian Groups ◽  
Finite Set ◽  
A Finite Group ◽  
Switching Class

AbstractA permutation group G on a finite set Ω is always exposable if whenever G stabilises a switching class of graphs on Ω, G fixes a graph in the switching class. Here we consider the problem: given a finite group G, which permutation representations of G are always exposable? We present solutions to the problem for (i) 2-generator abelian groups, (ii) all abelian groups in semiregular representations. (iii) generalised quaternion groups and (iv) some representations of the symmetric group Sn.

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].

scholarly journals Groups whose Chermak–Delgado lattice is a quasi-antichain

2019 ◽  
Vol 22 (3) ◽  
pp. 529-544
Author(s):  
Lijian An
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
A Finite Group

Abstract A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer {w\geq 3} , a quasi-antichain of width w is denoted by {\mathcal{M}_{w}} . In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that {\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if {w=1+p^{a}} for some positive integer a and some prime p. Let t be the number of abelian atoms in {\mathcal{CD}(G)} . In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?

scholarly journals On quasi-permutation representations of finite groups

1994 ◽  
Vol 36 (3) ◽  
pp. 301-308 ◽  
Author(s):  
J. M. Burns ◽  
B. Goldsmith ◽  
B. Hartley ◽  
R. Sandling
Keyword(s):  
Finite Group ◽  
Vector Space ◽  
Permutation Group ◽  
Minimal Degree ◽  
Permutation Representation ◽  
Permutation Groups ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Group Situation ◽  
A Finite Group

In [6], Wong defined a quasi-permutation group of degree n to be a finite group G of automorphisms of an n-dimensional complex vector space such that every element of G has non-negative integral trace. The terminology derives from the fact that if G is a finite group of permutations of a set ω of size n, and we think of G as acting on the complex vector space with basis ω, then the trace of an element g ∈ G is equal to the number of points of ω fixed by g. In [6] and [7], Wong studied the extent to which some facts about permutation groups generalize to the quasi-permutation group situation. Here we investigate further the analogy between permutation groups and quasipermutation groups by studying the relation between the minimal degree of a faithful permutation representation of a given finite group G and the minimal degree of a faithful quasi-permutation representation. We shall often prefer to work over the rational field rather than the complex field.

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].

Characteristics of GWrSn

1976 ◽  
Vol 79 (3) ◽  
pp. 433-441
Author(s):  
A. G. Williams
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Wreath Product ◽  
Generating Functions ◽  
Block Structure ◽  
Clifford Theory ◽  
A Finite Group ◽  
The Relationship

The ‘characteristics’ of the wreath product GWrSn, where G is a finite group, are certain polynomials (to be defined in section 2) which are generating functions for the simple characters of GWrSn. Schur (8) first used characteristics of the symmetric group. Specht (9) defined characteristics for GWrSn and found a relation between the characteristics of GWrSn and those of Sn which determined the simple characters of GWrSn. The object of this paper is to describe the p-block structure of GWrSn in the case where p is not a factor of the order of G. We use the relationship between the characteristics of GWrSn and those of Sn, which we deduce from a knowledge of the simple characters of GWrSn (these can be determined, independently of Specht's work, by using Clifford theory).

On the Frobenius spectrum of a finite group

2017 ◽  
Vol 16 (03) ◽  
pp. 1750051 ◽  
Author(s):  
Jiangtao Shi ◽  
Wei Meng ◽  
Cui Zhang
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Finite Groups ◽  
Prime Divisor ◽  
Complete Classification ◽  
Composition Factor ◽  
Frobenius Theorem ◽  
Even Order ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] any divisor of [Formula: see text], the order of [Formula: see text]. Let [Formula: see text], Frobenius’ theorem states that [Formula: see text] for some positive integer [Formula: see text]. We call [Formula: see text] a Frobenius quotient of [Formula: see text] for [Formula: see text]. Let [Formula: see text] be the set of all Frobenius quotients of [Formula: see text], we call [Formula: see text] the Frobenius spectrum of [Formula: see text]. In this paper, we give a complete classification of finite groups [Formula: see text] with [Formula: see text] for [Formula: see text] being the smallest prime divisor of [Formula: see text]. Moreover, let [Formula: see text] be a finite group of even order, [Formula: see text] the set of all Frobenius quotients of [Formula: see text] for even divisors of [Formula: see text] and [Formula: see text] the maximum Frobenius quotient in [Formula: see text], we prove that [Formula: see text] is always solvable if [Formula: see text] or [Formula: see text] and [Formula: see text] is not a composition factor of [Formula: see text].

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