The Maximum Order of an Element of a Finite Symmetric Group

1987 ◽  
Vol 94 (6) ◽  
pp. 497 ◽  
Author(s):  
William Miller
Keyword(s):  
Symmetric Group ◽  
Maximum Order ◽  
Finite Symmetric Group ◽  
Order Of An Element

scholarly journals Effective bounds for the maximal order of an element in the symmetric group

1989 ◽  
Vol 53 (188) ◽  
pp. 665-665 ◽  
Author(s):  
Jean-Pierre Massias ◽  
Jean-Louis Nicolas ◽  
Guy Robin
Keyword(s):  
Symmetric Group ◽  
Maximal Order ◽  
Order Of An Element

scholarly journals Affine symmetric group

2021 ◽  
Vol 4 (1) ◽  
pp. 3
Author(s):  
Joel Brewster Lewis
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Mathematical Structure ◽  
Number Line ◽  
Geometric Description ◽  
Generators And Relations ◽  
Higher Dimensional ◽  
Finite Set ◽  
Finite Symmetric Group

The affine symmetric group is a mathematical structure that describes the symmetries of the number line and the regular triangular tesselation of the plane, as well as related higher dimensional objects. It is an infinite extension of the symmetric group, which consists of all permutations (rearrangements) of a finite set. In additition to its geometric description, the affine symmetric group may be defined as the collection of permutations of the integers (..., −2, −1, 0, 1, 2, ...) that are periodic in a certain sense, or in purely algebraic terms as a group with certain generators and relations. These different definitions allow for the extension of many important properties of the finite symmetric group to the infinite setting, and are studied as part of the fields of combinatorics and representation theory.

scholarly journals Some Symmetric Identities Involving the Stirling Polynomials Under the Finite Symmetric Group

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 332
Author(s):  
Dongkyu Lim ◽  
Feng Qi
Keyword(s):  
Symmetric Group ◽  
Bernoulli Polynomials ◽  
Higher Order ◽  
Finite Symmetric Group ◽  
Symmetric Identities

In the paper, the authors present some symmetric identities involving the Stirling polynomials and higher order Bernoulli polynomials under all permutations in the finite symmetric group of degree n. These identities extend and generalize some known results.

On the ranks of certain finite semigroups of transformations

1987 ◽  
Vol 101 (3) ◽  
pp. 395-403 ◽  
Author(s):  
Gracinda M. S. Gomes ◽  
John M. Howie
Keyword(s):  
Symmetric Group ◽  
Transformation Semigroup ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Finite Semigroups ◽  
Idempotent Rank ◽  
Finite Symmetric Group ◽  
Rank 2 ◽  
Image Position

It is well-known (see [2]) that the finite symmetric group Sn has rank 2. Specifically, it is known that the cyclic permutationsgenerate Sn,. It easily follows (and has been observed by Vorob'ev [9]) that the full transformation semigroup on n (< ∞) symbols has rank 3, being generated by the two generators of Sn, together with an arbitrarily chosen element of defect 1. (See Clifford and Preston [1], example 1.1.10.) The rank of Singn, the semigroup of all singular self-maps of {1, …, n}, is harder to determine: in Section 2 it is shown to be ½n(n − 1) (for n ≽ 3). The semigroup Singn it is known to be generated by idempotents [4] and so it is possible to define the idempotent rank of Singn as the cardinality of the smallest possible set P of idempotents for which <F> = Singn. This is of course potentially greater than the rank, but in fact the two numbers turn out to be equal.

On the Determination of the Maximum Order of the Group of a Tournament

1973 ◽  
Vol 16 (1) ◽  
pp. 11-14 ◽  
Author(s):  
B. Alspach ◽  
J. L. Berggren
Keyword(s):  
Automorphism Group ◽  
Symmetric Group ◽  
Maximum Order ◽  
Odd Order

Let denote the automorphism group of the tournament T. Let g(n) be the maximum of taken over all tournaments of order n. It was noted in [3] that g(n) is also the order of the subgroups of Sn of maximum odd order where Sn denotes the symmetric group of degree n.

scholarly journals The Degree of the Splitting Field of a Random Polynomial over a Finite Field

10.37236/1823 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
John D. Dixon ◽  
Daniel Panario
Keyword(s):  
Finite Field ◽  
Symmetric Group ◽  
Random Permutation ◽  
Splitting Field ◽  
Random Polynomial ◽  
The Mean ◽  
Order Of An Element

The asymptotics of the order of a random permutation have been widely studied. P. Erdös and P. Turán proved that asymptotically the distribution of the logarithm of the order of an element in the symmetric group $S_{n}$ is normal with mean ${1\over2}(\log n)^{2}$ and variance ${1\over3}(\log n)^{3}$. More recently R. Stong has shown that the mean of the order is asymptotically $\exp(C\sqrt{n/\log n}+O(\sqrt{n}\log\log n/\log n))$ where $C=2.99047\ldots$. We prove similar results for the asymptotics of the degree of the splitting field of a random polynomial of degree $n$ over a finite field.

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