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.
28
questions
17
votes
0
answers
953
views
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 ...
32
votes
3
answers
3k
views
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 $...
12
votes
0
answers
537
views
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
$...
47
votes
1
answer
2k
views
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 ...
8
votes
1
answer
1k
views
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}, \...
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 ...
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 ...
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 ...
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 ...
17
votes
1
answer
1k
views
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 ...
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 ...
9
votes
1
answer
225
views
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\...
9
votes
1
answer
410
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 ...
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 ...
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\...
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 ...
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 ...
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 $486$ ...
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 ...
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\...
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 $B$.
...
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\...
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 ...
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 $\mathbb R$ ($\mathbb R \mathbb ...
2
votes
1
answer
188
views
Isomorphic Sylow p-subgroups of FSym$(\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 $C_p\wr C_p\wr\dots$ ...
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 ...
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 ...
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 \...