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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Groups generated by 3 involutions

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
Stefan Kohl's user avatar
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32 votes
3 answers
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Order of products of elements in symmetric groups

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order $...
Stefan Kohl's user avatar
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12 votes
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Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers

Definition / Question Definition: Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $...
Stefan Kohl's user avatar
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47 votes
1 answer
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Transitivity on $\mathbb{N}_0$ -- a 42 problem

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
Stefan Kohl's user avatar
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8 votes
1 answer
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Symplectic groups $Sp_{2m}(2)$ as $2$-transitive permutation (i.e. Galois) groups

I am looking for information about the symplectic groups $Sp_{2m}(2)$ as permutation group acting on quadratic forms. Consider the block matrices $$e=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \...
Maurizio Monge's user avatar
7 votes
0 answers
300 views

Does this class of groups contain finitely generated infinite periodic groups?

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
Stefan Kohl's user avatar
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2 votes
0 answers
260 views

Characterization of the elements of an infinite simple group

Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$. Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition $\tau_{r_1(m_1),r_2(m_2)}$ be the ...
Stefan Kohl's user avatar
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22 votes
4 answers
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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 ...
21 votes
4 answers
982 views

Is there a way of canonically labelling permutation groups?

When working with large numbers of graphs, a canonical labelling routine is essential as, after the initial cost of canonically labelling each graph, it permits isomorphism checks to be replaced with ...
Gordon Royle's user avatar
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17 votes
1 answer
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Does O'Nan-Scott depend on CFSG?

My question is in the title. Some context: there are two versions of the O'Nan-Scott theorem. The first, weaker version, is due to O'Nan and Scott (independently) and gives the structure of the ...
Nick Gill's user avatar
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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
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9 votes
1 answer
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Is $\beta\mathbb N$ a unique compactification with the smallest possible permutation group?

For a compactification $c\mathbb N$ of $\mathbb N$ let $\mathcal H(c\mathbb N,\mathbb N)$ be the group of homeomorphisms $h:c\mathbb N\to c\mathbb N$ such that $h(x)=x$ for all $x\in c\mathbb N\...
Taras Banakh's user avatar
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9 votes
1 answer
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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 ...
M. Winter's user avatar
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8 votes
2 answers
535 views

How hard is it to compute the diameter and the growth function of a finite permutation group of small degree?

Let $G \leq {\rm S}_n$ be a finite permutation group, and let $S = \{g_1, \dots, g_k\}$ be a generating set for $G$ which is closed under inversion and which does not contain the identity. The growth ...
Stefan Kohl's user avatar
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8 votes
2 answers
597 views

sum-sets in a finite field

Let $\mathbb{F}_p$ be a finite field, $A=\{a_1,\dots,a_k\}\subset\mathbb{F}_p^*$ a $k$-element set, for $k<p$. $\mathfrak{S}_k=$permutation gp. Question. Is it true there is always a $\pi\in\...
T. Amdeberhan's user avatar
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
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7 votes
2 answers
412 views

Automorphism group of a special commuting graph

Suppose $S_6$ is the symmetric group on six letters and let $X$ denote the conjugacy class containing $(12)(34)$. Define a graph $\Gamma$ with vertex set $X$ and edges precisely the 2-element subsets ...
maryam's user avatar
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6 votes
1 answer
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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 $486$ ...
Sebastien Palcoux's user avatar
6 votes
1 answer
422 views

Doubly primitive groups with simple socle

The classification of doubly transitive groups with simple socle is known. A good account of such classification can be found for example in this paper: Cameron, Peter J. Finite permutation groups ...
Leandro Vendramin's user avatar
6 votes
1 answer
157 views

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\...
Jens Bossaert's user avatar
3 votes
0 answers
299 views

What's the ratio of inclusions of finite groups with a distributive lattice?

Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$. ...
Sebastien Palcoux's user avatar
3 votes
1 answer
126 views

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\...
Jens Bossaert's user avatar
2 votes
0 answers
656 views

What is the definition of plethysm in the representation theory of permutation groups

Let $s_\lambda \circ s_\mu$ be a plethysm. Here let $\lambda, \mu$ be $m,n$ box Young diagrams. I have seen the definition of plethysms in symmetric functions. I would like to understand the ...
vishmay's user avatar
  • 349
2 votes
1 answer
731 views

Does Schur's Lemma hold in this case? Regular representations of $S_n$ over $\mathbb R$

Warning I am a physicist and I am not familiar with a lot of the machinary of representation theory. I consider the regular representation of $\mathbb S_n$ over reals $\mathbb R$ ($\mathbb R \mathbb ...
vishmay's user avatar
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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
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
1 vote
2 answers
188 views

Complexity of decision problem to decide if permutation group is $k$-transitive

Given a finite permutation group $G$ (a subgroup of the symmetric group on a finite set) in terms of its generators, what is known about the decision problem of deciding if $G$ is $k$-transitive for a ...
StefanH's user avatar
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0 votes
0 answers
140 views

Subgroups of powers of the alternating group on 5 elements

Definition: Let $G$ be a group, and let $H \leq G$ be a subgroup. We say that $H$ is big in $G$ if for every intermediate subgroup $H \leq L \leq G$ there exists some $x \in L$ such that $\langle H \...
Pablo's user avatar
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