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.

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What is known about $p$-local subgroups of Frobenius groups with (elementary abelian) kernel $F$ and cyclic complement $H$?

I'd like to ask the following question. What is known about $p$-local subgroups of Frobenius groups $G\cong F\rtimes H$ with (elementary abelian (if need be)) kernel $F$ and cyclic complement $H$? I ...
Bernhard Boehmler's user avatar
3 votes
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How can I generate all unique row/column permutations of a given incidence (binary) matrix using group theory and matrix multiplication?

Given a binary matrix, we need to generate all distinct matrices that are isomorphic to this input matrix under row and column permutations. I think this problem might involve some group theory rather ...
Gobind Puniani's user avatar
2 votes
1 answer
103 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
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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 ...
Fabio Mastrogiacomo's user avatar
8 votes
1 answer
171 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
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5 votes
1 answer
189 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
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1 vote
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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 ...
ccriscitiello's user avatar
2 votes
2 answers
194 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
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1 answer
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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}...
ABB's user avatar
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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 ...
THC's user avatar
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4 votes
2 answers
290 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
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4 votes
1 answer
167 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
224 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
110 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
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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, ...
Sophie MacDonald's user avatar
2 votes
1 answer
184 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
106 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
594 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
368 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
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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 $\{p_1,\dots,p_k\}\subseteq S_n$ ...
Max Alekseyev's user avatar
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1 answer
249 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
104 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
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0 answers
101 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
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6 votes
1 answer
229 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
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4 votes
1 answer
176 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
101 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
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0 votes
0 answers
138 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
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3 votes
1 answer
196 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
196 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
299 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
  • 11.4k
6 votes
1 answer
283 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
101 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
297 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\...
Hongyi Huang's user avatar
1 vote
2 answers
178 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
274 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
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11 votes
1 answer
240 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
  • 18.6k
7 votes
1 answer
201 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
448 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
  • 1,123
2 votes
1 answer
161 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
  • 3,871
20 votes
1 answer
592 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
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6 votes
2 answers
320 views

Irreducible factors of primitive permutation group representation

Consider a primitive permutation group $\Gamma\subseteq\mathrm{Sym}(N)$ on the $n$-element set $N=\{1,...,n\}$, that is, $\Gamma$ does not preserve any non-trivial partition of $N$. Consider the ...
M. Winter's user avatar
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6 votes
1 answer
2k views

Are there infinitely many insipid numbers?

A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ ...
Sebastien Palcoux's user avatar
4 votes
1 answer
359 views

Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S_\omega$?

Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S_\omega$, the group of permutations of the set of non-negative integers $\omega$?
Dominic van der Zypen's user avatar
9 votes
1 answer
357 views

Connections between linear representations and permutation representations

A finite group $\Gamma$ might be represented by a linear transformation $$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$ or by permutations $$\phi :\Gamma\to\mathrm{Sym}(n).$$ Of course, latter ones can ...
M. Winter's user avatar
  • 11.4k
3 votes
1 answer
152 views

Multiplication in $Z(\mathbb{C}S_n)$ [duplicate]

I am trying to multiply two generators of center $Z(\mathbb{C}[S_n])$ of ring algebra of symmetric group of $n$ elements. We know that these generators are given by sums of conjugacy classes in $S_n,$ ...
Filip's user avatar
  • 1,537
0 votes
2 answers
762 views

Generators for permutation groups

Consider (e.g.) the full permutation group $G=S_6$. A valid set of generators and equations for $G$ is $r^6=m^2=(rm)^5=1$. I say this system has width $3$ (because there are $3$ equations), length $10$...
Hauke Reddmann's user avatar
0 votes
1 answer
114 views

Can the permutation group of a matrix be generated by the subset of row permutation and column permutation?

Consider a 3*3 matrix $\left( \begin{array}{*{20}{c}} {{m_{11}}}&{{m_{12}}}&{{m_{13}}}\\ {{m_{21}}}&{{m_{22}}}&{{m_{23}}}\\ {{m_{31}}}&{{m_{32}}}&{{m_{33}}} \end{array} \right)...
user67451's user avatar
4 votes
1 answer
135 views

Diameter for permutations of bounded support

Let $S\subset \textrm{Sym}(n)$ be a set of permutations each of which is of bounded support, that is, each $\sigma\in S$ moves $O(1)$ elements of $\{1,2,\dotsc,n\}$. Let $\Gamma$ be the graph whose ...
H A Helfgott's user avatar
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4 votes
1 answer
162 views

Example of primitive permutation group with a regular suborbit and a non-faithful suborbit

I would like some examples of groups $G$ satisfying all of the following criteria: $G<Sym(n)$, the symmetric group on $n$ letters, and $G$ is primitive. $G$ has a regular suborbit, i.e. if $M$ is ...
Nick Gill's user avatar
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30 votes
2 answers
1k views

Does the symmetric group $S_{10}$ factor as a knit product of symmetric subgroups $S_6$ and $S_7$?

By knit product (alias: Zappa-Szép product), I mean a product $AB$ of subgroups for which $A\cap B=1$. In particular, note that neither subgroup is required to be normal, thus making this a ...
John McVey's user avatar
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