# 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.

175
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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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1
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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 $n$. ...

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### Function with $\text{Sym}_k$-orbit as fiber / forgetting ordering of vectors

Fix $n,k >0$. Is there a continuous function
$$ f: \oplus^k \mathbb{R}^n \longrightarrow \mathbb{R}^m$$
for some $m$ so that each fiber is essentially given by an orbit of the permutation group $\...

4
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1
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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 ...

2
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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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0
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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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3
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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 $...

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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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### 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 ...

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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,...

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### 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,...

2
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1
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143
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### 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 $\mathbb{P}^1(k)$, ...

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1
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### 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 ...

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### 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 ...

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1
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### 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 $486$ ...

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### 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 $\omega$?

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### 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 ...

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### 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 $S_n,$ ...

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### 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 $10$...

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### 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)...

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### 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 ...

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### 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 ...

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### 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 ...

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### Permutation groups with diameter $O(n \log n)$

I suspect that many permutation puzzles can be solved in $O(n \log n)$ moves, which has led me to the following question/conjecture:
Suppose that
1. $P_i$ for $i<k=O(1)$ are permutations on an $n$ ...

11
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1
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### Permutation groups having a regular cyclic subgroup and a conjectured algebra of characters

Let $G$ be a transitive permutation group of degree $d$ having a cyclic regular subgroup $K = \langle k \rangle \cong C_d$. Let $\pi(g) = |\mathrm{Fix}(g)|$ be the permutation character of $G$ and let
...

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### Number of permutations that are products of disjoint cycles of distinct length

What is the number of permutations $\pi\in S_n$ that are products of disjoint cycles of distinct length? What is the number of permutations that are products of disjoint cycles such that no more than $...

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### What is an upper limit of relative size of conjugacy class of the transitive finite group?

What is
$$ \limsup_{n\to\infty} \sup_{\deg(G)=n} \left( \frac{\max_{x\in G} \left|\operatorname{conj}(x)\right|}{\operatorname{ord}(G)}\right),$$
$G$ transitive permutation group?
And what are the ...

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### Enlarging a subdegree-finite "almost transitive" permutation group to a transitive one? (follow-up)

Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has finitely many orbits, and every point stabiliser $G_x$ has finite orbits.
Can we always find a permutation $\tau\in\...

3
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1
answer

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### Enlarging a subdegree-finite "almost transitive" permutation group to a transitive one?

Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has finitely many orbits, and every point stabiliser $G_x$ has finite orbits. Now consider a permutation $\tau\in\...