# 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.

194
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### Holt's Theorem on doubly transitive groups with $2$-central involutions fixing only one letter

In D. F. Holt, Transitive permutation groups in which an involution central in a Sylow 2-subgroup fixes a unique point, Proc. London Math. Soc. 37 (1978), 165–192, Derek Holt classifies finite doubly ...

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### What can be said about these tensor representations of $\mathrm{SL}(2)$?

Let $W = V \otimes \dots \otimes V$, the product of $n$ copies of $V = \mathbb{C}^2$. Let $G$ and $H$ be two subgroups of the symmetric group $S_n$ and let $\chi$ be a character of $G$.
Associated to $...

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### Does the permutohedron satisfy any minimal distortion property for graph metric vs Euclidean distance?

We can look on the permutohedron as a kind of "embedding" of the Cayley graph of $S_n$ to the Euclidean space. (That Cayley graph is constructed by the standard generators, i.e. ...

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### 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 ...

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### 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 $$\...

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### 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 ...

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### 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 ...

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### 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.
...

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0
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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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? ...

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### 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 $...

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### 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 $\sigma$?
$$1\leq k\leq m \Rightarrow \sigma(k)=2k , \quad \sigma(m+k)=2k-1$$

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### 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 ...

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### 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 ...

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### 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 $1$-...

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### 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 ...

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### 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 ...

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### 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}...

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### 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 ...

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### 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 $n$. ...

4
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### 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 ...

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### 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 $S$ ...

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### 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 ...

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### 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, ...

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### 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 $C_p\wr C_p\wr\dots$ ...

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### 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 $p$?...

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### 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,...

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### 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 $...

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### 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 $\{p_1,\dots,p_k\}\subseteq S_n$ ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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:
...

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### 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 ...

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1
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### 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 ...

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### 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?

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### 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 ...

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### 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\{...

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### 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 ...

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### 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\...

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### 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 $...

12
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### 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 $W = V\wedge V$
(...

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1
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### 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 ...

7
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1
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### 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,...