Graphs with symmetric automorphism group

In this paper we describe a construction for finite abstract chiral 4-polytopes with Schlafli type $$\{3,3,k\}$${3,3,k} (with tetrahedral facets) and with an alternating or symmetric group as automorphism group. We use it to prove that for all but finitely many $$n$$n, both $$A_n$$An and $$S_n$$Sn are the automorphism groups of such a polytope. We also show that the vertex-figures of the polytopes obtained from our construction are chiral.

The BNSR-invariants of a group G are a sequence Σ1(G)⊇Σ2(G)⊇⋯ of geometric invariants that reveal important information about finiteness properties of certain subgroups of G. We consider the symmetric automorphism group ΣAutn and pure symmetric automorphism group PΣAutn of the free group Fn and inspect their BNSR-invariants. We prove that for n≥2, all the “positive” and “negative” character classes of PΣAutn lie in Σn−2(PΣAutn)∖Σn−1(PΣAutn). We use this to prove that for n≥2, Σn−2(ΣAutn) equals the full character sphere S0 of ΣAutn but Σn−1(ΣAutn) is empty, so in particular the commutator subgroup ΣAutn' is of type Fn−2 but not Fn−1. Our techniques involve applying Morse theory to the complex of symmetric marked cactus graphs.

We give a new approach to description of p-Sylow subgroup normalizes in the groups Sn (symmetric group on n letters).

This paper surveys some results in the area of maximal subgroups of the finite simple groups and their automorphism groups. The first two sections are concerned with maximal subgroups of the alternating and symmetric groups. We outline the Reduction Theorem and discuss the maximality of primitive groups in the corresponding symmetric groups. In the third section we consider the corresponding questions for the classical groups. The next section contains a brief survey of knowledge of the maximal subgroups of the other almost simple groups. Finally, the last section outlines some recent work on primitive permutation groups of special degrees.

The Schrödinger equation of an N‐electron closed‐shell system is reduced to that for the spatial wave function Ψ by the aid of the theory of the symmetric (permutation) group SN. The first‐order perturbation equation based on the Hartree–Fock‐SCF model as the zero‐order solution is then solved by decomposing it into a set of soluble 1/2(N/2)[(N/2) + 1] equations for the pair‐correlation (first‐order) spatial wave functions. These spatial functions give the spatial parts of the pair functions in the spin orbital basis formulation, reducing also the number of independent pairs needed in the first order.

Polytopes of high rank for the symmetric groups

Galatius’ most striking result is easy enough to state. Let Σn be the symmetric group on n letters and Fn be the free (non-abelian) group with n generators. The symmetric group Σn acts naturally by permutation on the n generators of Fn, and every permutation gives thus rise to an automorphism of the free group. Galatius proves that the map Σn → AutFn in homology induces an isomorphism in degrees less than (n− 1)/2. The homology of the symmetric groups in these ranges is well understood. In particular, in common with all finite groups, it has no non-trivial rational homology. By Galatius’ theorem, in low degrees this then is also true for AutFn: H∗(AutFn)⊗Q = 0 for 0 < ∗ < 2n/3,

The structure of the centralizers of finite subgroups in homogenous symmetric groups are given up to isomorphism by Güven-Kegel-Kuzucuoğlu in 2015 and Kuzucuoğlu-Oliynyk-Sushchansky in 2018. In this article, we answer the natural question; what is the structure of the normalizers of finite semiregular subgroups in homogenous symmetric groups? We prove, for some sequence We answer negatively, the following open question: does every automorphism of preserve a level? We show that, there exist uncountably many automorphisms of which does not preserve any level. Finally, we show that the level preserving automorphisms of the homogenous finitary symmetric group is isomorphic to

Let G ≀ SN be the wreath product of a finite group G and the symmetric group SN. The aim of this paper is to prove the branching theorem for the increasing sequence of finite groups G ≀ S1 ⊂ G ≀ S2 ⊂ ... ⊂ G ≀ SN ⊂ ... and the analog of Young's orthogonal form for this case, using the inductive approach invented by A. Vershik and A. Okounkov for the case of symmetric group.

Let x 1 x 2…xi xi+1 …xn be a permutation of {1,2,…,n} written in one line notation. An "i-adjacent interchange" applied to this permutation produces the permutation x 1 x 2…xi+1 xi …xn obtained from the first permutation by interchanging the symbols in positions i and i+1. We denote such an interchange by (i,i+1), where i=1,…,n−1. The permutation x 1 x 2…xi+1 xi …xn results from x 1 x 2…xi xi+1 …xn by right multiplication by the transposition (i,i+1) in the symmetric group Sn . Two adjacent transpositions (i,i+1) and (j,j+1) are themselves adjacent if either i=j+1 or j=i+1. In this paper, we show that that for every n≥3, there is an n long sequence (f(1),f(1)+1),…,(f(n!),f(n!)+1) of adjacent transpositions, each except the first adjacent to its predecessor, the first adjacent to the last, and such that the set of products of transpositions {πt:πt=(f(1),f(1)+1)…(f(t),f(t)+1)t=1,…,n!} is Sn , with the final product πn equal to the identity. We call such a sequence of transpositions a doubly adjacent Gray code for the symmetric group Sn . We give a procedure for constructing such a Gray code. The existence of such a code (Theorem 6.2) seems to be a very nontrivial problem. As a consequence of the existence of this code we resolve an open problem in the theory of Cayley graphs of the symmetric group by showing that the Cayley graph of Sn with generators {(l,2),(l,…,n),(n,…,l)} has a Hamiltonian cyclen≥3 (Corollary 6.5). Our procedure shows how to braid n strands.

The question of identifying the elements of the center of a nearring and of determining when that center is a subnearring is an area of continued research. We consider the centers of centralizer nearrings, $M_I(S_n)$, determined by the symmetric groups $S_n$ with $n geq 3$ and the inner automorphisms $I = Inn S_n$. General tools for determining elements of the center of $M_I(S_n)$ are developed, and we use these to list the specific elements in the centers of $M_I(S_4)$, $M_I(S_5)$, and $M_I(S_6)$.

It is the purpose of this paper to introduce a novel approach to clustering rank data on a set of possibly large cardinality n ∈ N*, relying upon Fourier representation of functions defined on the symmetric group Sn. In the present setup, covering a wide variety of practical situations, rank data are viewed as distributions on Sn. Cluster analysis aims at segmenting data into homogeneous subgroups, hopefully very dissimilar in a certain sense. Whereas considering dissimilarity measures/distances between distributions on the non commutative group Sn, in a coordinate manner by viewing it as embedded in the set [0, 1]n! for instance, hardly yields interpretable results and leads to face obvious computational issues, evaluating the closeness of groups of permutations in the Fourier domain may be much easier in contrast. Indeed, in a wide variety of situations, a few well-chosen Fourier (matrix) coefficients may permit to approximate efficiently two distributions on Sn as well as their degree of dissimilarity, while describing global properties in an interpretable fashion. Following in the footsteps of recent advances in automatic feature selection in the context of unsupervised learning, we propose to cast the task of clustering rankings in terms of optimization of a criterion that can be expressed in the Fourier domain in a simple manner. The effectiveness of the method proposed is illustrated by numerical experiments based on artificial and real data.

A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (where \(\mathcal {R}\) is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.

Representations on the cohomology of [formula omitted

We consider the problem of computing rearrangement distance of every permutation in the symmetric group Sn and present a simple algorithm for doing it. By analysing the rearrangement distance distribution computed for different scenarios, we were able to correct the reversal distance distribution given by Kececioglu and Sankoff; disprove a conjecture of Walter, Dias and Meidanis on signed reversal and transposition diameter; and reinforce the conjecture of Dias and Meidanis on prefix transposition diameter. As an attempt to better characterize how rearrangement distances are distributed, two new measures are introduced: the traversal diameter and the longevity. We conjecture results on them and on the diameter of some rearrangement distances.