Finite classical groups and multiplication groups of loops

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

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

Journal of Group Theory ◽

10.1515/jgth-2020-0058 ◽

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.

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A note on powers in simple groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500419600165x ◽

1997 ◽

Vol 122 (1) ◽

pp. 91-94 ◽

Cited By ~ 16

Author(s):

JAN SAXL ◽

JOHN S. WILSON

Keyword(s):

Conjugacy Class ◽

Simple Group ◽

Simple Groups ◽

Finite Simple Groups ◽

Classical Groups ◽

Similar Combination

In [7], the second author proved that there is an integer k such that every element of a finite non-abelian simple group S is a product of k commutators in S. The motivation for proving this result came from a model-theoretic question about simple groups. The proof depended on the classification of the finite simple groups, a theorem of Malle, Saxl and Weigel [5] which shows that in many finite simple classical groups S there is a real conjugacy class R such that S=R3∪{1}, and an ultraproduct argument. Here we shall use a similar combination of ideas to prove the following result.

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ON INVOLUTIONS AND INDICATORS OF FINITE ORTHOGONAL GROUPS

Journal of the Australian Mathematical Society ◽

10.1017/s144678871800006x ◽

2018 ◽

Vol 105 (3) ◽

pp. 380-416 ◽

Cited By ~ 1

Author(s):

GREGORY K. TAYLOR ◽

C. RYAN VINROOT

Keyword(s):

Asymptotic Behavior ◽

Simple Group ◽

Generating Functions ◽

Finite Simple Group ◽

Point Of View ◽

Character Degree ◽

Simple Groups ◽

Finite Simple Groups ◽

Classical Groups ◽

Orthogonal Groups

We study the numbers of involutions and their relation to Frobenius–Schur indicators in the groups $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius–Schur indicators are 1), and we observe this holds for all finite simple groups $G$ other than the groups $\unicode[STIX]{x1D6FA}^{\pm }(4m,q)$ with $q$ even. We prove computationally that for small $m$ this statement indeed holds for these groups by equating their character degree sums with the number of involutions. We also prove a result on a certain twisted indicator for the groups $\text{SO}^{\pm }(4m+2,q)$ with $q$ odd. Our second motivation is to continue the work of Fulman, Guralnick, and Stanton on generating functions and asymptotics for involutions in classical groups. We extend their work by finding generating functions for the numbers of involutions in $\text{SO}^{\pm }(n,q)$ and $\unicode[STIX]{x1D6FA}^{\pm }(n,q)$ for all $q$, and we use these to compute the asymptotic behavior for the number of involutions in these groups when $q$ is fixed and $n$ grows.

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On the involution fixity of simple groups

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091521000237 ◽

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.

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Elements with Square Roots in Finite Groups

Algebra Colloquium ◽

10.1142/s1005386705000647 ◽

2005 ◽

Vol 12 (04) ◽

pp. 677-690 ◽

Cited By ~ 5

Author(s):

M. S. Lucido ◽

M. R. Pournaki

Keyword(s):

Finite Group ◽

Finite Groups ◽

Simple Groups ◽

Finite Simple Groups ◽

Square Root ◽

Alternating Groups ◽

Square Roots ◽

A Finite Group ◽

Groups Of Lie Type

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

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