Questions tagged [permutation-groups]

For groups represented as permutations. Group transitivity, rank 3 groups, orbits and suborbits, stablizers, permutation characters, primitivity are all on-topic.

Filter by
Sorted by
Tagged with
2 votes
0 answers
79 views

Are the alternating pentultimate and the $2$-alternating pentultimate isomorphic?

Consider a physical puzzle which is in the shape of a dodecahedron where the pieces that move are the face centers and you can slice it through a plane parallel to two opposite faces going through the ...
Bram Cohen's user avatar
3 votes
1 answer
262 views

Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?

Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation. The corresponding Hurwitz representation $V_{\text{Hur}}$ has character $$\...
utx7563yu's user avatar
22 votes
4 answers
2k views

Open problems which might benefit from computational experiments

Question: I wonder what are the open problems , where computational experiments might me helpful? (Setting some bounds, excluding some cases, shaping some expectations ). Grant program: The context of ...
1 vote
0 answers
152 views

Diameter of the "Masterball-puzzle" permutation groups by a kind of Cartier-Foata enumeration?

There is an wonderful blog post by Jordan S. Ellenberg SHOULD YOU BE SURPRISED BY THE DIAMETER OF THE NXNXN RUBIK’S GROUP?. Which explains how one can come to $N^2log(N)$ estimate of the diameter of ...
Alexander Chervov's user avatar
7 votes
1 answer
374 views

Does every transitive permutation group contain a permutation whose cycle lengths have a common divisor?

Let $H$ be a transitive subgroup of $\mathfrak{S}_n$, $n \geq 2$. Using Jordan's lemma ($H$ is not a union of conjugate proper subgroups), we see that $H$ contains a permutation without fixed points. ...
Béranger Seguin's user avatar
1 vote
0 answers
78 views

cycle types of all words in a permutation group

I have been working with permutation groups. For a given $G\subset S_n$, what I have been computing depends only on the conjugacy class of $G$. Say all permutation groups in this question are ...
Pierre's user avatar
  • 2,145
8 votes
1 answer
590 views

A question regarding symmetrizing the tensor product of vectors in two different ways

Let $V = \mathbb{C}^m$, endowed with the standard hermitian inner product which we will denote by $\langle \cdot, \cdot \rangle$, $n$ be a positive integer and $\Sigma_n$ denote the symmetric group on ...
Malkoun's user avatar
  • 4,981
5 votes
1 answer
270 views

Questions about algorithms for permutation groups

Let $G < S_n$ be a permutation group of degree $n$, $\mathcal{P(n)}$ denote the set of all partitions of $n$, and $c: G \rightarrow \mathcal{P}(n)$, where $c(g)$ is the partition given by the ...
Victor Miller's user avatar
3 votes
0 answers
78 views

Diameters of permutation groups with transitive generators

Suppose that for a permutation puzzle on $n$ elements we have a subroutine to place any $m>3$ elements in arbitrary positions (possibly scrambling the rest). Can we solve the puzzle (if it is ...
Dmytro Taranovsky's user avatar
1 vote
0 answers
83 views

Totally imprimitive groups

Following [Neumann P.M., The lawlessness of groups of finitary permutation groups, Arch. Math. 55(6) 1990, 521-532], we define totally imprimitive groups in a more general form as follows: Let $G$ be ...
IGT's user avatar
  • 69
2 votes
0 answers
91 views

Conjugacy classes of $P\Gamma L(2,q)$

$\DeclareMathOperator\PGaL{P\Gamma L}\DeclareMathOperator\GF{GF}$May I know whether there are any developments made on the conjugacy classes of $\PGaL(2,q)$ where $q$ is a prime power but not a prime? ...
Chong Eu Meng's user avatar
12 votes
1 answer
430 views

abelian quotients of permutation groups

Let $G$ be a subgroup of the permutation group $S_n$, and let $H$ be a normal subgroup of $G$ such that the quotient group $G/H$ is abelian. What is the best known upper estimate for the cardinality $...
Yuri Bilu's user avatar
  • 1,130
2 votes
1 answer
112 views

If $n=2m$, what is the order of the permutation $\sigma(k)=2k , \quad \sigma(m+k)=2k-1$

Let $n=2m$. What is the order of the following permutation $\sigma$? $$1\leq k\leq m \Rightarrow \sigma(k)=2k , \quad \sigma(m+k)=2k-1$$
ABB's user avatar
  • 3,898
3 votes
0 answers
155 views

Centralizer of each element of a subgroup contained in the normalizer of the subgroup

Let $G \leq \operatorname{Sym}(\Omega)$ be a permutational group. A base for $G$ is a sequence of elements of $\Omega$ whose pointwise stabilizer is the identity. A base is called irredundant if no ...
Fabio Mastrogiacomo's user avatar
8 votes
1 answer
195 views

Asymptotic number of permutation representations of a given group

Let $G$ be a finitely generated group. I am trying to count the number of permutation representations on $n$ elements, i.e. homomorphisms from $G$ to the symmetric group $S_n$. Equivalently this is ...
Squala's user avatar
  • 964
5 votes
1 answer
220 views

Irreducible deleted permutation module for a finite group

Let $G$ be a subgroup of the symmetric group $S_n = \operatorname{Sym}(X)$. Let $k$ be a field, and let $V$ be the permutation module corresponding to $X$. Then $V$ is not irreducible, it has a $1$-...
spin's user avatar
  • 2,781
1 vote
0 answers
28 views

Adjacent vertices of a permutohedron which is the orbit of a point $x$ with repeated coordinates

Question: Let $x=(x_1, ..., x_n) \in \mathbb{R}^n$, and let $P$ be the convex hull of the points formed by permuting the coordinates of $x$. Given a vertex $y$ of $P$, what is a general rule for ...
ccriscitiello's user avatar
2 votes
2 answers
229 views

Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?

The background: We recall/define the following: $\Omega_n=\{1,\dots,n\}$. $M_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups ...
John McVey's user avatar
  • 1,018
0 votes
1 answer
211 views

A particular permutation on $\mathbb{Z}_n$

Let us consider $\mathbb{Z}_N=\{0,1,\cdots N-1\}$. Does there exist any permutation $\sigma$ on $\mathbb{Z}_N$ satisfying : $$\exists p,q\in\mathbb{N}, ~~p,q\neq0 (mod~N) ~~\forall i,j\in \mathbb{Z}...
ABB's user avatar
  • 3,898
1 vote
1 answer
153 views

Number of orbits for abelian group actions

Suppose $G$ is an abelian group acting faithfully on two sets, $X$ and $Y$, of the same size. None of $G$, $X$ and $Y$ is finite. Now suppose $G$ is the union of abelian groups $G_i$, where $i$ varies ...
THC's user avatar
  • 4,313
4 votes
2 answers
337 views

Minimal degree of primitive permutation group

Helmut Wielandt discussed an old question (Chap. 2, Section 15, which can be dated back to Camille Jordan): Let $g\neq 1$ be a permutation in some finite primitive permutation group $G$ of degree $n$. ...
Y. Zhao's user avatar
  • 3,317
4 votes
1 answer
175 views

Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$

I have been doing research on the Niemeier lattices with root systems of type, $A_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups ...
Sean Miller's user avatar
2 votes
1 answer
248 views

On Sylow subgroups of finitary symmetric groups

$\DeclareMathOperator\FSym{FSym}$Let $p$ be a prime and $S$ be a transitive Sylow $p$-subgroup of $\FSym(\mathbb{N})$, the finitary symmetric group of the set of all natural numbers. Question: Is $S$ ...
Ahmet Arikan's user avatar
2 votes
0 answers
126 views

The number of orbits of a two-point stabilizer of the symplectic group $Sp(2m,2)$

I am trying to figure out the number of orbits of a two-point stabilizer of the action of $Sp(2m,2)$ on its two orbits $\Omega^+$ and $\Omega^-$ as detailed in Dixon and Mortimer's "Permutation ...
F.Tomas's user avatar
  • 121
1 vote
0 answers
55 views

Effect on finite transformation semigroup under a particular modification of the generators

The following question arises in connection with problems in automata theory related to the road problem. Let $f_1, f_2: [N] \to [N]$ be maps such that the transformation semigroup $S = \langle f_1, ...
Sophie M's user avatar
  • 675
2 votes
1 answer
188 views

Isomorphic Sylow p-subgroups of FSym$(\mathbb{N})$

Let FSym$(\mathbb{N})$ denote the finitary symmetric group on the natural numbers. Are all Sylow p-subgroups of FSym$(\mathbb{N})$ isomorphic (maybe to the iterated wreath product $C_p\wr C_p\wr\dots$ ...
Ahmet Arikan's user avatar
0 votes
1 answer
110 views

Sylow $p$-subgroups of FSym($\mathbb N$)

$\DeclareMathOperator\FSym{FSym}$Let $\FSym(\mathbb{N})$ denote the finitary symmetric group on the set of natural numbers. How many Sylow $p$-subgroups does $\FSym(\mathbb{N})$ have for any prime $p$?...
Ahmet Arikan's user avatar
16 votes
2 answers
619 views

A sum over partitions involving "subpartitions"

Consider the following sum over partitions of $n$: $$ S(n)=\sum_{\substack {j_1,\dots,j_n\geq 0\\j_1+2j_2+\dots+nj_n=n}} \prod_{t=1}^n \frac{1}{j_t!t^{j_t}}f_t(j_1,\dots,j_t),$$ where $$ f_t(j_1,\dots,...
Christian Bertoni's user avatar
5 votes
3 answers
457 views

Generation of permutation groups by fixed elements subgroups

Suppose $(H,X)$ is a permutation group (with $H$ a group acting faithfully on the set $X$). Under what circumstances is $H$ generated by its subgroups $H_x$, where $H_x$ is the subgroup of $H$ fixing $...
THC's user avatar
  • 4,313
1 vote
0 answers
35 views

Uniform cover of the symmetric group by "multiples" of its subset

Given a subset $T$ of the symmetric group $S_n$. For $p\in S_n$, define $$pT = \{ pt\mid t\in T\}.$$ Questions: Q1: Is there a simple characterization of all subsets $\{p_1,\dots,p_k\}\subseteq S_n$ ...
Max Alekseyev's user avatar
0 votes
1 answer
274 views

Given the index of two permutations, Is there a direct way to compute the index of their composition? [closed]

Suppose you are given two indexed permutations, (7 followed by 4, for instance) What is the best way to go about finding their composition, given the indices themselves? I'd imagine that the answer ...
Sean Miller's user avatar
2 votes
0 answers
124 views

Permutation group with a nice lattice of block systems

Let $X$ be a finite set and $G$ be a transitive subgroup of the symmetric group on $X.$ Recall that a (complete) block system for this action is a partition of $X = B_1 \cup \cdots \cup B_k$ into ...
tim's user avatar
  • 388
0 votes
0 answers
102 views

classification for some groups

Let $G$ be a finite group. Suppose that $G$ acts on a set, say $X$, transitively such that for every $x\in X$, $G_x^g=G_x$ or $G_x^g\cap G_x=\{1\}$. Could you please tell me if there is a ...
salam's user avatar
  • 1
6 votes
1 answer
243 views

Is there a known classification of regular multiplicity-free permutation groups?

The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$. $\Sigma$ is regular if it acts ...
M. Winter's user avatar
  • 12.5k
4 votes
1 answer
182 views

Number of permutations with combinatorial geometric constraints

We are given a $d$-dimensional hypercube $H$, where each vertex is labeled with an integer $\ell\in\{1, 2, \ldots, 2^d\}$. Let $L$ be this labelling. Question: How many labelling permutations $L'$ of ...
Penelope Benenati's user avatar
2 votes
1 answer
122 views

Source for: a permutation group is multiplicity-free if and only if its 2-orbits define an association scheme

I have recently proven the following (at least, so I believe): Theorem. Given a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on the set $\Omega:=\{1,...,n\}$, the following are equivalent: ...
M. Winter's user avatar
  • 12.5k
0 votes
0 answers
163 views

Request for a modern Reference for Frobenius' paper "Über die Charaktere der mehrfach transitiven Gruppen"

I'm interested in the paper of Jan Saxl "The Complex Characters of the Symmetric Groups that Remain Irreducible in Subgroups". I have only (not yet enough!) standard background on the ...
gualterio's user avatar
  • 1,043
3 votes
1 answer
203 views

Does the Hawaiian Earring Group embed into the permutation group of $\mathbb N$?

Recall that the Hawaiian earring group, $\mathbb G$, is the fundamental group of the Hawaiian Earing using the point at the origin. It can be understood more combinatorially as a subgroup of the ...
Corey Bacal Switzer's user avatar
7 votes
1 answer
209 views

Automorphism group of a putative strongly regular graph

The smallest feasible parameters for which no strongly regular graph is known to exist are $(69,48,32,36)$, as per Brouwer's table. What is known on the automorphism group of such a graph?
Patrick Sole's user avatar
7 votes
1 answer
325 views

For which $n$ can $S_n$ act transitively on $n+k$ elements?

It is known that the symmetric group $S_n$ can act transitively on $n+1$ elements if and only if $n=5$. Are there similar classifications for $S_n$ acting transitively on $n+k$ elements, where $k$ is ...
M. Winter's user avatar
  • 12.5k
6 votes
1 answer
285 views

Group action with unique word

This must be known or easy for some of you, but here goes: Suppose $f_0,f_1:[n]\to [n]$ are invertible functions, where $[n]=\{0,\dots,n-1\}$ is a set of $n$ elements. For a word $w=w_1\dots w_m\in\{...
Bjørn Kjos-Hanssen's user avatar
7 votes
0 answers
108 views

Extensions of oligomorphic groups

Recall that a permutation group $G\le\mathfrak{S}(\mathbb N)$ is oligomorphic if, for any positive integer $k$, the induced $G$-action on $k$-tuples of distinct elements of $\mathbb N$ has only ...
Yves Stalder's user avatar
9 votes
2 answers
313 views

Can $1\ne H\cap H^g\lhd H$ happen if $G$ is a primitive permutation group with stabiliser $H$?

Assume everything is finite. Let $G$ be a primitive permutation group with point stabiliser $G_\alpha$ for some $\alpha$. For $\beta\ne\alpha$, by an arc stabiliser we mean $G_{\alpha\beta}=G_\alpha\...
Groups's user avatar
  • 369
1 vote
2 answers
188 views

Is there a 2 transitive finite group with transitive normal subgroup having a cyclic quotient other than $A_n$ and $S_n$?

Let $G \leq S_n$ be $2$-transitive other than $A_n$ and $S_n$. Is it possible that there exists $N\lhd G$ with $N\neq G$, $N$ transitive and $G/N$ cyclic? I am interested mostly in the answer when $...
Lior Bary-Soroker's user avatar
12 votes
0 answers
276 views

How many steps on $S_n$ are required to span $V\wedge V$, $V = K^n$?

Let $A$ be a set of generators of $G=S_n$; assume $e\in A$, $A=A^{-1}$. Let $V = K^n$, $K$ a field. Consider the natural action of $G$ on $V$ (namely, $g(e_i) = e_{g(i)}$) and on $W = V\wedge V$ (...
H A Helfgott's user avatar
  • 19.3k
11 votes
1 answer
244 views

How many steps are required for double transitivity?

Let $A$ be a set of generators of $S_n$, or of a doubly transitive subgroup of $S_n$. Assume $e\in A$, $A=A^{-1}$. What is the least $k$ such that $A^k$ is doubly transitive as a set? That is, what is ...
H A Helfgott's user avatar
  • 19.3k
7 votes
1 answer
224 views

Permutation groups generated by finitely many point stabilisers

Assume that $G\leq\operatorname{Sym}(X)$ is a permutation group generated by all its point stabilisers, i.e. $G=\langle G_x \mid x\in X\rangle$. There is no cardinality restriction on $X$. Furthermore,...
Jens Bossaert's user avatar
7 votes
1 answer
521 views

Wreath product $S_k\wr S_n$ inside $S_{kn}$

I want to understand wreath products a little better. Currently, my intuition about them is as follows. Take $nk$ disks, pile them in order forming $n$ piles of $k$ disks (one pile contains disks {1,...
thedude's user avatar
  • 1,417
2 votes
1 answer
172 views

Non-commutative projective lines

There have been many approaches to the notion of projective line: combinatorial approaches (e.g. as certain permutation groups, such as $\mathrm{PGL}_2(k)$ in its natural action on $\mathbb{P}^1(k)$, ...
THC's user avatar
  • 4,313
20 votes
1 answer
610 views

If $S_\mathbb N$ is partitioned into finitely many pieces, must one piece contain a "skew copy" of every countable group?

Suppose $G$ and $H$ are groups. $C \subseteq H$ is called a skew copy of $G$ in $H$ if $C = hK$ for some $h \in H$ and some subgroup $K$ of $H$ with $K \cong G$. Question 1: Suppose the infinite ...
Will Brian's user avatar
  • 17.4k