scholarly journals On the involution fixity of simple groups

2021 ◽  
pp. 1-19
Author(s):  
Timothy C. Burness ◽  
Elisa Covato
Keyword(s):  
Fixed Points ◽  
Permutation Group ◽  
Simple Groups ◽  
Classical Groups ◽  
Exceptional Group ◽  
Sporadic Group ◽  
Group Of Lie Type ◽  
Primitive Groups ◽  
Finite Permutation Group

Abstract Let $G$ be a finite permutation group of degree $n$ and let ${\rm ifix}(G)$ be the involution fixity of $G$ , which is the maximum number of fixed points of an involution. In this paper, we study the involution fixity of almost simple primitive groups whose socle $T$ is an alternating or sporadic group; our main result classifies the groups of this form with ${\rm ifix}(T) \leqslant n^{4/9}$ . This builds on earlier work of Burness and Thomas, who studied the case where $T$ is an exceptional group of Lie type, and it strengthens the bound ${\rm ifix}(T) > n^{1/6}$ (with prescribed exceptions), which was proved by Liebeck and Shalev in 2015. A similar result for classical groups will be established in a sequel.

scholarly journals A characterisation of almost simple groups with socle 2E6(2) or M(22)

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Chris Parker ◽  
M. Reza Salarian ◽  
Gernot Stroth
Keyword(s):  
Simple Group ◽  
Simple Groups ◽  
Exceptional Group ◽  
Sporadic Simple Group ◽  
Group Of Lie Type ◽  
Almost Simple Groups

AbstractWe show that the sporadic simple group M(22), the exceptional group of Lie type

Finite classical groups and multiplication groups of loops

1995 ◽  
Vol 117 (3) ◽  
pp. 425-429 ◽  
Author(s):  
Ari Vesanen
Keyword(s):  
Permutation Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Classical Groups ◽  
Alternating Groups ◽  
Multiplication Group ◽  
Finite Classical Groups ◽  
Left And Right

Let Q be a loop; then the left and right translations La(x) = ax and Ra(x) = xa are permutations of Q. The permutation group M(Q) = 〈La, Ra | a ε Q〉 is called the multiplication group of Q; it is well known that the structure of M(Q) reflects strongly the structure of Q (cf. [1] and [8], for example). It is thus an interesting question, which groups can be represented as multiplication groups of loops. In particular, it seems important to classify the finite simple groups that are multiplication groups of loops. In [3] it was proved that the alternating groups An are multiplication groups of loops, whenever n ≥ 6; in this paper we consider the finite classical groups and prove the following theorems

scholarly journals On the minimal degree of a transitive permutation group with stabilizer a 2-group

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Primož Potočnik ◽  
Pablo Spiga
Keyword(s):  
Fixed Points ◽  
Permutation Group ◽  
Minimal Degree ◽  
Simple Groups ◽  
Transitive Permutation Group ◽  
Finite Simple Groups ◽  
Trivial Element ◽  
Stabilizer Of A Point

AbstractThe minimal degree of a permutation group 𝐺 is defined as the minimal number of non-fixed points of a non-trivial element of 𝐺. In this paper, we show that if 𝐺 is a transitive permutation group of degree 𝑛 having no non-trivial normal 2-subgroups such that the stabilizer of a point is a 2-group, then the minimal degree of 𝐺 is at least \frac{2}{3}n. The proof depends on the classification of finite simple groups.

scholarly journals ON THE HEIGHT AND RELATIONAL COMPLEXITY OF A FINITE PERMUTATION GROUP

2021 ◽  
pp. 1-40
Author(s):  
NICK GILL ◽  
BIANCA LODÀ ◽  
PABLO SPIGA
Keyword(s):  
Permutation Group ◽  
Model Theory ◽  
Independent Set ◽  
Maximum Size ◽  
Proper Subset ◽  
Permutation Groups ◽  
Finite Permutation Group ◽  
Pointwise Stabilizer ◽  
Relational Complexity

Abstract Let G be a permutation group on a set $\Omega $ of size t. We say that $\Lambda \subseteq \Omega $ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $\Lambda $ . We define the height of G to be the maximum size of an independent set, and we denote this quantity $\textrm{H}(G)$ . In this paper, we study $\textrm{H}(G)$ for the case when G is primitive. Our main result asserts that either $\textrm{H}(G)< 9\log t$ or else G is in a particular well-studied family (the primitive large–base groups). An immediate corollary of this result is a characterization of primitive permutation groups with large relational complexity, the latter quantity being a statistic introduced by Cherlin in his study of the model theory of permutation groups. We also study $\textrm{I}(G)$ , the maximum length of an irredundant base of G, in which case we prove that if G is primitive, then either $\textrm{I}(G)<7\log t$ or else, again, G is in a particular family (which includes the primitive large–base groups as well as some others).

Linear Symmetry Classes

1976 ◽  
Vol 28 (6) ◽  
pp. 1311-1319 ◽  
Author(s):  
L. J. Cummings ◽  
R. W. Robinson
Keyword(s):  
Vector Space ◽  
Permutation Group ◽  
Cyclic Group ◽  
Symmetry Class ◽  
Complex Vector Space ◽  
Complex Vector ◽  
Linear Character ◽  
Finite Permutation Group ◽  
Symmetry Classes ◽  
Finite Dimensional

A formula is derived for the dimension of a symmetry class of tensors (over a finite dimensional complex vector space) associated with an arbitrary finite permutation group G and a linear character of x of G. This generalizes a result of the first author [3] which solved the problem in case G is a cyclic group.

scholarly journals Intransitive Permutation Groups with Bounded Movement Having Maximum Degree

2019 ◽  
Vol 16 ◽  
pp. 8272-8279
Author(s):  
Behnam Razzagh
Keyword(s):  
Positive Integer ◽  
Fixed Points ◽  
Permutation Group ◽  
Maximum Degree ◽  
Permutation Groups ◽  
Ams Classification ◽  
Bounded Movement

Let G be a permutation group on a set with no fixed points in and let m be a positive integer. If for each subset of  the size  is bounded, for , we define the movement of g as the max  over all subsets of . In this paper we classified all of permutation groups on set of size 3m + 1 with 2 orbits such that has movement m . 2000 AMS classification subjects: 20B25

Modeling questions for quantum permutations

2018 ◽  
Vol 21 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Teodor Banica ◽  
Amaury Freslon
Keyword(s):  
Permutation Group ◽  
Matrix Model ◽  
Classical Groups

Given a quantum permutation group [Formula: see text], with orbits having the same size [Formula: see text], we construct a universal matrix model [Formula: see text], having the property that the images of the standard coordinates [Formula: see text] are projections of rank [Formula: see text]. Our conjecture is that this model is inner faithful under suitable algebraic assumptions, and is in addition stationary under suitable analytic assumptions. We prove this conjecture for the classical groups, and for several key families of group duals.

Casimir operators of the exceptional group G2 in a B3 basis

1994 ◽  
Vol 72 (7-8) ◽  
pp. 319-320
Author(s):  
Adam M. Bincer
Keyword(s):  
Classical Groups ◽  
Exceptional Group ◽  
Casimir Operators

Expressions are given for the Casimir operators of the exceptional group G2 in a concise form similar to that used for the classical groups. This is achieved by expressing the generators of G2 in a B3 basis.

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