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.
48
questions with no upvoted or accepted answers
17
votes
0
answers
955
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 ...
12
votes
0
answers
276
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 $W = V\wedge V$
(...
12
votes
0
answers
699
views
Solving a set of equations in a finite symmetric group
A standard way to find solutions to a finite set of equations in a finite symmetric group
${\rm S}_n$ is to take the equations as relators of a finitely presented group, to use
the low index subgroups ...
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
$...
9
votes
0
answers
186
views
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 $n$ ...
8
votes
0
answers
226
views
Connection between Gauss's lemma and Zolotarev's lemma
So I was reflecting on the relationship between Gauss's Lemma and Zolotarev's Lemma in proofs of quadratic reciprocity:
GL: $(a/p) = -1^n$, where $n$ is the number of least positive residues of $ax$ ...
7
votes
0
answers
108
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 ...
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 ...
5
votes
0
answers
196
views
Application of finding shortest paths on Cayley graphs
For a fixed integer number $m$, Consider Cayley graph defined by all m-cycles in Symmetric group $Sym(n)$.
I know that for $m=2$,
there are some applications of finding shortest paths (or distance ...
5
votes
0
answers
287
views
Uniqueness of the direct product decomposition of inclusions of finite groups
This post is a generalization of Uniqueness of the direct product decomposition of finite groups.
Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups.
Definition: ...
5
votes
0
answers
180
views
Information about permutation character from local action
Let $G$ be a finite permutation group acting transitively, but not regularly, on a set $V$. Let $H$ be the stabilizer of some point $v\in V$, and suppose that $H$ acts 2-transitively on one of its (...
5
votes
0
answers
236
views
Finite groups generated by 3 involutions interchanging disjoint residue classes of the integers
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 ...
4
votes
0
answers
89
views
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. ...
4
votes
0
answers
160
views
Young Tableau Box Correlations
Let $T$ be a uniformly random Standard Young Tableau (SYT) of shape $\lambda=(\lambda_1,\cdots,\lambda_k)$ with $|\lambda|=n$. Let $T_{ij}$ denote the value in box $(i,j)$. I'm interested in what can ...
4
votes
0
answers
236
views
Permutations with all cycles odd
I have recently needed to compute the number of permutations in $S_n$ with all cycles odd, and while this is easy using Flajolet-Sedgewick theory (the exponential generating function seems to be $\...
4
votes
0
answers
250
views
A conjecture on Zassenhaus groups
In a 1995 paper by Ali Nesin, "Permutation groups of finite Morley rank", the following conjecture is mentioned:
Conjecture 5 An infinite Zassenhaus group $G$ of finite Morley rank is isomorphic ...
3
votes
0
answers
78
views
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 ...
3
votes
0
answers
156
views
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 ...
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
0
answers
110
views
Double coset relation for unique intermediate subgroup (with homogeneity)
Let $G$ be a group and $H$ a subgroup such that there is a unique (non-trivial) intermediate subgroup $K$ (i.e. $H < S < G$ implies $S=K$) and $(H \subset K) \sim (K \subset G)$ (homogeneity) ...
3
votes
0
answers
156
views
A section from subfactors to transitive groups
A finite group-subgroup subfactor is a subfactor $(N \subset M)$ isomorphic to $(R^G \subset R^H)$ with $(H \subset G)$ an inclusion of finite groups acting as outer automorphism on the hyperfinite II$...
3
votes
0
answers
172
views
Largest permutation groups without "non-mixing" subgroups
We say that a subgroup of ${\rm Sym}(\mathbb{N})$ has sparse orbit representatives
if it has infinitely many orbits on $\mathbb{N}$, but the set of smallest orbit
representatives has natural density 0 ...
2
votes
0
answers
84
views
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 ...
2
votes
0
answers
91
views
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? ...
2
votes
0
answers
128
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 ...
2
votes
0
answers
124
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 ...
2
votes
0
answers
76
views
First appearance of "structure tree"?
Let $G$ be a transitive permutation group acting on a set $\Omega$. A structure tree $T$ for $(G,\Omega)$ is defined as follows: if $G$ is primitive, then it consists of a root node connected by edges ...
2
votes
0
answers
65
views
What are the transitive extensions of finite representations of cyclic groups?
This question is a generalisation of this one. Let $H$ be a finite, transitive permutation group of degree $n$. If the point stabiliser subgroup $H_n$ of degree $n-1$ is some faithful permutation ...
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
0
answers
156
views
Special sets of involutions generating ${\rm S}_n$
For which positive integers $k$ and $r$ are there involutions $g_{n,i} \in {\rm S}_n$
$(n \in \mathbb{N}, \ i = 1, \dots, k)$ such that the following hold?:
for any $n$, the $g_{n,i}$ $(i = 1, \dots, ...
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 ...
1
vote
0
answers
79
views
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 ...
1
vote
0
answers
83
views
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 ...
1
vote
0
answers
28
views
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 ...
1
vote
0
answers
55
views
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, ...
1
vote
0
answers
35
views
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 $\{p_1,\dots,p_k\}\subseteq S_n$ ...
1
vote
0
answers
90
views
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 ...
1
vote
0
answers
269
views
A question about permutation matrices
This question is trying to abstract out in a self-contained way the point that is probably being made in page 6 of this paper, https://arxiv.org/pdf/1604.03544.pdf and why Theorem 4.1 there works.
...
1
vote
0
answers
239
views
Counting elements having a given cycle structure in maximal subgroups of a generalized symmetric group
Let $G$ be the wreath product $C_7\wr S_{18}$, where $C_7$ is the cyclic group of order 7 and $S_{18}$ is the symmetric group on 18 symbols. Consider $G$ to be embedded in the group $S_{126}=S_{7\cdot ...
1
vote
0
answers
124
views
graphs with semiregular automorphisms
I need some "well-known" non-regular finite graphs (at least two vertices have different valency) whose automorphism groups contain a non-trivial subgroup that acts on the vertices semi-regularly (i....
1
vote
0
answers
51
views
Set of vectors closed under restricted permutation operations
Let $V=\{v_1,\cdots v_k\}$ a set of non zero different norm one vectors in $R^d$, $k>2$. I am trying to demonstrate that if $Q=\{P_i\in R^{k\times k},i=1,\cdots,k\}$ is a set of permutations such ...
1
vote
0
answers
179
views
Are the finite groups inclusions, almost all relatively cyclic?
Definition: An inclusion of finite groups $(A \subset B)$ is relatively cyclic if $\exists b \in B$ such that $\langle A,b \rangle = B$.
Definition: Two inclusions of finite groups are equivalent, $(...
1
vote
0
answers
219
views
A connection between nonplanar complete graphs and the alternating groups?
I didn't get any response on MSE so I though I'd give this a try here (my question on MSE).
I went to an undergrad's senior honors thesis presentation a while ago. She was discussing crossing numbers ...
1
vote
0
answers
220
views
Outer automorphisms 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 ...
0
votes
0
answers
102
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 ...
0
votes
0
answers
163
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 ...
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 \...
0
votes
0
answers
217
views
Orbits of stabilizer of two points in a 2-transitive permutation group
I was doing something which needs to know sizes of all orbits of the stabilizer of two points in a 2-transitive permutation group. Since all 2-transitive permutation groups are known ans so are their ...