All Questions
Tagged with finite-groups permutation-groups
81 questions
0
votes
1
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65
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Groups with $2$-transitive permutation representations of different degrees
Suppose $G$ is a finite group, and suppose that it acts $2$-transitively in each of the permutation representations $(G,X_i)$ ($i$ ranges over some index set $I$), where the $X_i$s all have different ...
- 4,547
4
votes
0
answers
115
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Complexity to find "short" (e.g. polynomial in diameter) decomposition of the permutation into the product of generators?
Question 1: Consider the symmetric group $S_n$ and some set of permutations $p_i$. Given permutation $g$ - what is known about the algorithmic complexity to decompose $g$ into product of $p_i$ ...
- 24.9k
3
votes
1
answer
162
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Does a finite non-abelian Group $G$ have a primitive nonabelian quotient $G/N$?
Let $G$ be a finite nonabelian group which is transitive with degree $d$. It is understood that we may construct a primitive group $H = P \wr G$ where $P$ is primitive such that $H/P^d = G$. Now I'm ...
1
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0
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65
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Doubly transitive groups in which a one point stabilizer has a normal subgroup of even size
In 1972, Hering classified the finite doubly transitive permutation groups $(G,X)$ ($G$ acting faithfully on $X$) in which $G_x$, with $x \in X$, contains a normal subgroup $N_x$ of even order which ...
- 4,547
1
vote
0
answers
94
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Sylow subgroups of doubly transitive groups
Let $(G,X)$ be a doubly transitive permutation group (where $G$ acts faithfully on the set $X$). Let $x \in X$, and suppose that $\vert X \vert = n + 1$ is finite. Now let $p$ be a prime divisor of $n$...
- 4,547
2
votes
0
answers
174
views
Doubly transitive groups in which a point stabilizer has an abelian normal subgroup
Let $G$ be a finite doubly transitive group in its action on the set $X$, such that a point stabilizer $G_x$ ($x \in X$) has an abelian normal subgroup $N_x$.
I have read that if $\vert N_x \vert$ is ...
- 4,547
3
votes
1
answer
129
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Holt's Theorem on doubly transitive groups with $2$-central involutions fixing only one letter
In D. F. Holt, Transitive permutation groups in which an involution central in a Sylow 2-subgroup fixes a unique point, Proc. London Math. Soc. 37 (1978), 165–192, Derek Holt classifies finite doubly ...
- 4,547
2
votes
0
answers
101
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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 ...
- 31
3
votes
1
answer
321
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Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?
Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation.
The corresponding Hurwitz representation $V_{\text{Hur}}$ has character $$\...
- 175
4
votes
1
answer
251
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Diameter of the "Masterball-puzzle" permutation groups by a kind of Cartier-Foata enumeration?
There is an wonderful blog post by Jordan S. Ellenberg SHOULD YOU BE SURPRISED BY THE DIAMETER OF THE NXNXN RUBIK’S GROUP?. Which explains how one can come to $N^2log(N)$ estimate of the diameter of ...
- 24.9k
4
votes
0
answers
88
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Diameters of permutation groups with transitive generators
Suppose that for a permutation puzzle on $n$ elements we have a subroutine to place any $m>3$ elements in arbitrary positions (possibly scrambling the rest). Can we solve the puzzle (if it is ...
- 7,545
12
votes
1
answer
450
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abelian quotients of permutation groups
Let $G$ be a subgroup of the permutation group $S_n$, and let $H$ be a normal subgroup of $G$ such that the quotient group $G/H$ is abelian. What is the best known upper estimate for the cardinality $...
- 1,294
2
votes
1
answer
114
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If $n=2m$, what is the order of the permutation $\sigma(k)=2k , \quad \sigma(m+k)=2k-1$
Let $n=2m$. What is the order of the following permutation $\sigma$?
$$1\leq k\leq m \Rightarrow \sigma(k)=2k , \quad \sigma(m+k)=2k-1$$
- 4,058
5
votes
1
answer
253
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Irreducible deleted permutation module for a finite group
Let $G$ be a subgroup of the symmetric group $S_n = \operatorname{Sym}(X)$.
Let $k$ be a field, and let $V$ be the permutation module corresponding to $X$.
Then $V$ is not irreducible, it has a $1$-...
- 2,821
4
votes
3
answers
328
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Is a point stabilizer in the Mathieu group $M_{20}$ half-transitive?
The background: We recall/define the following:
$\Omega_n=\{1,\dots,n\}$.
$M_n$ is the Mathieu group of degree $n$. We follow the Wikipedia article "Mathieu group" and define these groups ...
- 1,068
0
votes
1
answer
213
views
A particular permutation on $\mathbb{Z}_n$
Let us consider $\mathbb{Z}_N=\{0,1,\cdots N-1\}$.
Does there exist any permutation $\sigma$ on $\mathbb{Z}_N$ satisfying :
$$\exists p,q\in\mathbb{N}, ~~p,q\neq0 (mod~N) ~~\forall i,j\in \mathbb{Z}...
- 4,058
4
votes
2
answers
369
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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 $n$. ...
- 3,337
4
votes
1
answer
194
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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 ...
- 113
2
votes
0
answers
150
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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 ...
- 121
5
votes
3
answers
498
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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 $...
- 4,547
7
votes
1
answer
344
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 ...
- 13.6k
9
votes
2
answers
329
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\...
- 379
1
vote
2
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193
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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 $...
- 3,362
11
votes
1
answer
248
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 ...
- 20.2k
7
votes
1
answer
582
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,...
- 1,549
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 $486$ ...
1
vote
2
answers
1k
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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 $10$...
- 4,829
4
votes
1
answer
152
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 ...
- 20.2k
4
votes
1
answer
202
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 ...
- 11.2k
31
votes
2
answers
1k
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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 ...
- 1,068
10
votes
0
answers
194
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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 $n$ ...
- 7,545
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 ...
- 119
3
votes
1
answer
506
views
A permutation with reflection property
Consider permutations $\pi$ of the set $\{1,\dots,n\}$ having the symmetry property $\pi \pi^* \pi = \pi^*$, where $\pi^*$ is the "reflection" $k \mapsto n+1-k$. Are there references or other ...
- 1,848
3
votes
2
answers
338
views
Length of composition series in a primitive group
Let $G$ be a primitive group acting on a set $\Omega$ with $n$ elements. By Cameron/Liebeck (essentially a consequence of the Classification + O'Nan-Scott), there are two possibilities:
(a) $G$ has a ...
- 20.2k
5
votes
1
answer
502
views
Time Complexity of the Word Problem for Finite Permutation Groups
Given a finite permutation group, i.e. a subgroup of the symmetric group on $n$ symbols in terms of generators, what is the complexity of the word problem? That is, computing if two words in the ...
- 798
1
vote
1
answer
80
views
Complexity to decide for permutation group if every element fixed at most $k$ points
I want to consider the following problem, which generalises the decision problem to decide if a given finite permutation group is a Frobenius group:
Given a finite permutation group in terms of its ...
- 798
1
vote
2
answers
207
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 ...
- 798
20
votes
3
answers
940
views
What did Frobenius prove about $M_{12}$?
I am interested in this paper which I can't read because it's in German:
Frobenius, G., Über die Charaktere der mehrfach transitiven Gruppen., Berl. Ber. 1904, 558-571 (1904). ZBL35.0154.02.
A free ...
- 11.2k
14
votes
1
answer
959
views
The number of involutions in a permutation group
If $G$ is a group let $I(G)$ be the number of involutions (elements of order 2) in $G$. My question is then easily stated: does there exists a constant $C > 1$ such that for every $n \ge 1$ and ...
- 3,399
16
votes
2
answers
2k
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How do I know if an irreducible representation is a permutation representation?
I have a vague question, a less vague question and a lot of vaguer questions about permutation representations of a finite group $G$.
Vague question. Recall that if $G$ acts on a finite set $X$, we ...
- 575
1
vote
0
answers
242
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 ...
- 1,021
1
vote
1
answer
286
views
Automorphism group of a graph
Suppose $\Gamma$ is a simple graph and $G=\mathrm{Aut}(\Gamma)$ is the automorphism group of $\Gamma$. If $G$ stabilizes a subgraph $\Gamma_1$,, and $G_0$ is the point-wise stabiliser of the set $V(\...
- 81
7
votes
2
answers
422
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 ...
- 81
16
votes
1
answer
1k
views
Tensor power of the natural representation of Sn
The symmetric group $S_n$ acts over $V=\mathbb{R}^n$ by permuting the canonical basis.
So it acts over $V^{\otimes p}$ with a diagonal action (acts the same over each element of the tensor product).
...
- 583
5
votes
1
answer
193
views
composition factors of primitive components
A finite transitive permutation group $G$ can always be ``decomposed'' into primitive permutation groups, called its primitive components, although the decomposition is not unique. See Chapter 1 of ...
- 537
8
votes
2
answers
617
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\...
- 43.2k
1
vote
0
answers
54
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 ...
- 329
2
votes
1
answer
964
views
Is there a formula for the number of elements in $S_n$ having length $k$ with respect to the generators taken to be the transpositions?
Define the length of a permutation as the minimum number of adjacent transpositions needed to describe it.
We know that there is only one element of length $0$ in $S_n$ and $n-1$ elements of length $1$...
- 21
7
votes
6
answers
1k
views
Transitive permutation groups which all of their proper subgroups are intransitive
Let $G$ be a transitive permutation group on a finite set $\Omega$. It is clear
that if $G$ is regular, then every proper subgroup of $G$ is intransitive. Is
there any other class of groups with this ...
- 792
2
votes
1
answer
137
views
Intransitive finite irreducible linear groups whose orbits are all large
I am interested in intransitive irreducible linear subgroups $G\subseteq\mathrm{GL}_n(\mathbb{F}_p)$ acting on $V-\{0\}=\mathbb{F}_p^n-\{0\}$ in the natural way, such that all of the orbits are very ...
- 537