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1
vote
2answers
213 views

Does the hyperoctahedral group have only 3 maximal normal subgroups?

An hyperoctahedral group $G$ is the wreath product of $S_2$ and $S_n$, where $S_{n}$ is the symmetric group on $n$ letters, or in other words the semi-direct product $G=S_2^n\rtimes S_n$, w.r.t. the ...
12
votes
0answers
450 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 $...
23
votes
2answers
2k 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 ...
6
votes
2answers
326 views

Can every permutation group be realized as the automorphism group of a graph (acting on a subset of the vertices)?

By Frucht's theorem, every finite group can be realized as the automorphism group of a finite undirected graph. Because a permutation group is a finite group, it is clear that every permutation group ...
4
votes
0answers
200 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
2answers
285 views

Classification of generously transitive groups

A permutation group $G \lt S_n$ is called generously transitive, if for each $i,j$ there exists a permutation that interchanges them. Is there a reasonable classification of such (finite) groups?
2
votes
1answer
187 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 $1$...
5
votes
6answers
476 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 ...
2
votes
1answer
171 views

2-closure of a permutation group

Let $G$ be a group acting on a set $\Omega$ faithfully. Then 2-closure of $G$ denoted by $G^{(2)}$ is the largest subgroup of the symmetric group of $\Omega$ with the same orbits as $G$ on $\Omega\...
5
votes
1answer
789 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}, \...
2
votes
1answer
103 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 ...
5
votes
1answer
189 views

Hyperoctahedral group acting on a special permutation

Let $[n]=\{1,...,n\}$ and $[\hat n]=\{\hat 1,...,\hat n\}$. Realize the hyperoctahedral group $H_n$ as the centralizer of the permutation $(1\hat 1)\cdots (n \hat n)$. It has $2^n n!$ elements. Let $...
5
votes
1answer
110 views

A sum over characters of $S_{2n}$ and zonal spherical functions of $(S_{2n},H_n)$

The hyperoctahedral group $H_n$ can be seen as the centralizer of the permutation $(12)(34)\cdots (2n-1\,2n)$ in $S_{2n}$. It has $2^nn!$ elements. The quantities $$ \omega_\lambda(\pi)=\frac{1}{2^...
-1
votes
1answer
122 views

Can someone explain how Sims's algorithm works on a permutation group with a simple example? [closed]

Can someone explain how Sims's algorithm works on a permutation group with a simple example? The book "Permutation group algorithms" by Seress is a pretty hard read with a whole bunch of confusing ...
12
votes
1answer
287 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 $\...
15
votes
2answers
447 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 ...
6
votes
1answer
256 views

number of maximal subgroups of the symmetric group

What is the asymptotics of the number of the maximal subgroups of $S_n$ (as a function of $n$)? This must be written down somewhere... EDIT I am actually more interested in the number of conjugacy ...
0
votes
1answer
166 views

Reference Request: Irreducibles of the regular representation of the permutation group is absolutely irreducible

I am writing a paper(physics) where I am using the fact that the irreducible's of the regular representations of the permutation group are absolutely irreducible in the following sense. If $V$ is an ...
5
votes
2answers
190 views

Group acting on two different sets, can isomorphisms be computed efficiently?

For the permutation group isomorphism problem, a permutational isomorphism between two permutation groups is sought. A closely related but seemingly easier problem arises, if an isomorphism between ...
5
votes
1answer
116 views

What is the criterion for a matrix containing vectors and their permutations being invertible?

Consider the matrix $A\in\mathbb{R}^{m\times 2m}$. Let any arbitrary choice of $m$ columns of $A$ be linearly independent. Together with a permutation $P\in\mathcal{P_{2m}}$, one can build the matrix $...
42
votes
1answer
1k 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 ...
3
votes
1answer
207 views

variance of the number of fixed points for a permutation group

It is reasonably well-known that the variance of the number of fixed points for $S_n$ equals $1.$ Now, what about other transitive permutation groups on $\{1, \dotsc, n\}?$ Presumably much is known. I ...
4
votes
0answers
127 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 ...
1
vote
0answers
176 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 ...
1
vote
0answers
169 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, $(...
0
votes
0answers
114 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 \...
4
votes
0answers
129 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 ...
20
votes
3answers
673 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 ...
15
votes
2answers
430 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....
7
votes
1answer
118 views

Covering a set with images of a transversal

Let $G$ be a permutation group on a finite set $\Omega$ with orbits $\Omega_1,\ldots,\Omega_k$. By a transversal we mean a set $\lbrace\omega_1,\ldots,\omega_k\rbrace$ with $\omega_j\in\Omega_j$ for ...
10
votes
0answers
346 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 ...
3
votes
2answers
259 views

If d(“G/H”) < d(G) = 2, must H contain a primitive element?

Let $G$ be a finite group that can be generated by $2$ elements, and let $H \leq G$ be a (not necessarily normal) subgroup for which there exists some $g \in G$ such that $H \langle g\rangle = G$. ...
6
votes
1answer
496 views

Permutation Group Question

A question about permutation groups: I wonder if someone who is expert in permutation group theory could answer the following question. Let $x \in S_n$ (the symmetric group) be an involution which ...
8
votes
1answer
386 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 ...
9
votes
2answers
267 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 ...
3
votes
3answers
274 views

Mclaughlin Graph

how can i construct a strongly regular graph with parameter $(275,112,30,56)$(Mclaughlin Graph), (105,32,4,12)? I need adjacency matrix of them? I know they are unique.
2
votes
0answers
128 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
0answers
145 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$...
4
votes
1answer
406 views

Generalization of a property of $A_n; n\geq 5$

Let $H$ and $K$ be two proper non-trivial subgroups of the alternating group $A_n; n\geq 5$. Then there exists a maximal subgroup $M$ of $A_n$ such that $H\not\leq M$ and $K\not\leq M$. To see this ...
10
votes
4answers
2k views

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 elements, ...
1
vote
1answer
236 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 ...
5
votes
1answer
288 views

Minimum word length for an unusual set of generators of the symmetric group

Problem. Let $n\geq 2$ and let $T$ be the set of all permutations in $S_n$ of the form $$t_k:=\prod_{1\leq i\leq k/2}(i,k-i) \qquad \hbox{for $k=3,4,\ldots,n+1$}.$$ Find the least integer $f_n$...
3
votes
0answers
275 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 $B$. ...
5
votes
0answers
109 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: ...
2
votes
1answer
212 views

Classification of indecomposable inclusions $(H \subset G)$ with $G$ decomposable

Definition: A group $G$ is indecomposable if: $G = G_1 \times G_2 \Rightarrow \exists i \ G_i = 1$. We can generalize the notion of indecomposable from groups to inclusion of groups as ...
6
votes
1answer
316 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 ...
2
votes
1answer
314 views

When is the product of two permutations cyclic?

Call a subset $H \subset S_n$ "simple" if for every $q \in S_n$, there is some $p \in H$ such that $pq$ is cyclic (i.e. consists of a single cycle of length $n$). Is there some characterization known ...
4
votes
0answers
143 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 ...
27
votes
3answers
1k views

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 ...
2
votes
1answer
159 views

simple graphs of degree 16 with a semiregular normal subgroup isomorphic to the quaternion group $Q_8$

Is there any simple graph $\Gamma$ with 16 vertices with full automorphism group $G$ such that $H\cong Q_8$ be a semiregular normal subgroup of $G$?