Affine 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.

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

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.

POSITIVE PERMUTATION BRAIDS AND PERMUTATION INVERSIONS WITH SOME APPLICATIONS

In this paper we constructed an isomorphic group of binary matrices to a finite symmetric group. Our method is based on the inversion of permutations. Using this embedding we find an algorithm for writing down a standard braid word representation for each positive permutation braid. Also an algorithm for writing basis of Hecke algebra Hn+1 from such basis of Hn is given.

Transitive simple subgroups of wreath products in product action

A transitive simple subgroup of a finite symmetric group is very rarely contained in a full wreath product in product action. All such simple permutation groups are determined in this paper. This remarkable conclusion is reached after a definition and detailed examination of ‘Cartesian decompositions’ of the permuted set, relating them to certain ‘Cartesian systems of subgroups’. These concepts, and the bijective connections between them, are explored in greater generality, with specific future applications in mind.

Permutations with Restricted Cycle Structure and an Algorithmic Application

Let q be an integer with q [ges ] 2. We give a new proof of a result of Erdös and Turán determining the proportion of elements of the finite symmetric group Sn having no cycle of length a multiple of q. We then extend our methods to the more difficult case of obtaining the proportion of such elements in the finite alternating group An. In both cases, we derive an asymptotic formula with error term for the above mentioned proportion, which contains an unexpected occurrence of the Gamma-function.We apply these results to estimate the proportion of elements of order 2f in Sn, and of order 3f in An and Sn, where gcd(2, f) = 1, and gcd(3, f) = 1, respectively, and log f is polylogarithmic in n. We also give estimates for the probability that the fth power of such elements is a transposition or a 3-cycle, respectively. An algorithmic application of these results to computing in An or Sn, given as a black-box group with an order oracle, is discussed.

On the ranks of certain finite semigroups of transformations

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.