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.
191
questions
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 ...
32
votes
3
answers
3k
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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 $...
30
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 ...
28
votes
3
answers
2k
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When is $S_n \times S_m$ a subgroup of $S_p$?
I asked the following question on math.stackexchange several months ago:
Let $n,m,p>1$ be such that $S_n \times S_m \hookrightarrow S_p$. Does it imply that $p \geq n+m$?
Derek Holt gave a ...
22
votes
4
answers
2k
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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 ...
20
votes
3
answers
917
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 ...
20
votes
1
answer
610
views
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 ...
19
votes
6
answers
4k
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An easy proof that $S(n)$ does not embed into $A(n+1)$?
Rotman's book An Introduction to the Theory of Groups (Fourth Edition) asks, on page 22, Exercise 2.8, to show that $S(n)$ cannot be embedded in $A(n+1)$, where $S(n)$ = the symmetric group on $n$ ...
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 ...
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 ...
16
votes
2
answers
701
views
Minimal maximal subgroup of the symmetric group
The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we ...
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 ...
16
votes
2
answers
619
views
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,...
16
votes
1
answer
451
views
Escaping from a centralizer
Let $G = Sym(n)$, $n$ even. Let $H<G$ be the stabilizer of the partition $\{\{1,2\},\{3,4\},\dotsc,\{n-1,n\}\}$, or, what is the same, the centralizer of $(1\;2) \dotsc (n-1\; n)$.
By Stirling's ...
15
votes
2
answers
668
views
Explicit permutation representation of the Thompson sporadic simple group?
The Thompson group Th of order $90745943887872000$ is one of the sporadic simple groups occurring in the classification of finite simple groups.
Its maximal subgroups are known (see http://brauer....
14
votes
1
answer
976
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).
...
14
votes
1
answer
877
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 ...
13
votes
1
answer
944
views
Outer automorphism action on representations of $S_6$
Let $S_6$ be the symmetric group on 6 letters and let $\alpha \colon S_6 \to S_6$ be an outer automorphism (note that $S_6$ is the only permutation group that has an outer automorphism and that $\...
12
votes
1
answer
430
views
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 $...
12
votes
1
answer
661
views
A sum over characters of the symmetric group
Let $C_\mu$ be the size of the conjugacy class in $S_n$ of permutations whose cycletype is the partition $\mu\vdash n$. Let $\chi$ be the characters of the irreducible representations of $S_n$.
Let $\...
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
$...
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 ...
11
votes
1
answer
281
views
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
...
10
votes
1
answer
569
views
Maximal subgroups of a certain finite 2-group
The following came up in a problem on reconstruction of digraphs. I determined enough about the answer to satisfy the application completely, but still I am curious to know what the complete solution ...
10
votes
1
answer
245
views
sum of character product over derangements
It is widely known that
$$ \frac{1}{n!}\sum_{\pi\in S_n}\chi_\lambda(\pi)\chi_\mu(\pi)=\delta_{\lambda,\mu},$$
where $S_n$ is the permutation group and $\chi$ are its irreducible characters.
In ...
10
votes
1
answer
2k
views
Permutation character of the symmetric group on subsets of certain size
The symmetric group $S_n$ acts on $[n]:=\{1,\ldots,n\}$, thereby inducing an action on the set $$\wp_k(n)=\{\: A\subseteq[n] \::\: \#A=k \:\}$$ of subsets of cardinality $k$, simply by $$(g,A)\mapsto ...
9
votes
1
answer
555
views
Generalization of Frobenius groups
Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point.
In other words, if in a ...
9
votes
2
answers
313
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\...
9
votes
2
answers
788
views
Regular elementary abelian subgroups of primitive permutation groups
A finite group $B$ is said to be a B-group if every primitive permutation group having a regular (transitive) subgroup isomorphic to B is $2$-transitive.
Schur proved that a cyclic group of ...
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 ...
9
votes
2
answers
421
views
Vertex-primitive graphs with two vertices having almost the same neighbourhood
Hypothesis: Let $\Gamma$ be a vertex-primitive graph with two vertices $u$ and $v$ such that $$|N(u) \cap N(v)|=|N(v)|-1$$
Question: Is it true that $\Gamma$ must either be a complete graph or have ...
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
463
views
Question about doubly transitive groups with an n-cycle
Let $G$ be a doubly transitive subgroup of $S_n$ which contains an $n$-cycle, and let $G_{12}$ be the subgroup of $G$ consisting of all elements $g\in G$ for which $g(1)=1$ and $g(2)=2$. Then $G_{12}$...
9
votes
0
answers
186
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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 $n$ ...
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
3
answers
678
views
partly obscured Rubik's cube
I just came back from a beach which features a large Rubik's cube (2m high). The base of the cube is not visible and the top is not coloured. The four vertical sides are each divided $3\times 3$ into ...
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}, \...
8
votes
1
answer
652
views
Center of one-point stabilizer in 2-transitive groups
In this MO question it was mentioned that the following fact seems to be true:
If $G$ is doubly transitive on $X$ and the one-point stabilizer $G_x$ has a
non-trivial center, then $G$ is of ...
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
1
answer
195
views
Asymptotic number of permutation representations of a given group
Let $G$ be a finitely generated group.
I am trying to count the number of permutation representations on $n$ elements, i.e. homomorphisms from $G$ to the symmetric group $S_n$.
Equivalently this is ...
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 ...
8
votes
1
answer
462
views
Permutations of prescribed cycle types that multiply to the identity
Suppose that $\lambda_1,\lambda_2,\lambda_3$ are partititions of $n$. When do there exist permutations $\sigma_1,\sigma_2,\sigma_3 \in S_n$ such that
(1) $\sigma_1\sigma_2\sigma_3$ is the identity;
...
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
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 ...
7
votes
1
answer
498
views
Linear permutations commuting with $x\rightarrow x^{-1}$
Let $F = \operatorname{GF}(2^n)$ be a finite field. Define a permutation $\phi:F \rightarrow F$ by the formula
$$
\phi(x) = x^{-1}, \ x\neq 0; \ \phi(0) =0.
$$
We say that a permutations $\psi$ of $F$ ...
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 ...
7
votes
1
answer
375
views
Does every transitive permutation group contain a permutation whose cycle lengths have a common divisor?
Let $H$ be a transitive subgroup of $\mathfrak{S}_n$, $n \geq 2$.
Using Jordan's lemma ($H$ is not a union of conjugate proper subgroups), we see that $H$ contains a permutation without fixed points.
...