# 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.

150 questions
Filter by
Sorted by
Tagged with
38 views

264 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 $W = V\wedge V$ (...
228 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 ...
145 views

### Permutation groups generated by finitely many point stabilisers

Assume that $G\leq\operatorname{Sym}(X)$ is a permutation group generated by all its point stabilisers, i.e. $G=\langle G_x \mid x\in X\rangle$. There is no cardinality restriction on $X$. Furthermore,...
116 views

### Is there exists a lattice isomorphism?

Let $\text{P}$ be the set of partitions of {1,2,...,n} and $\text{Y}$ the set of Young subsets of permutation group S(n)(the coxeter group of type An). As is well-known, the set $\text{Y}$ is a ...
293 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,...
110 views

### Non-commutative projective lines

There have been many approaches to the notion of projective line: combinatorial approaches (e.g. as certain permutation groups, such as $\mathrm{PGL}_2(k)$ in its natural action on $\mathbb{P}^1(k)$, ...
496 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 ...
189 views

### Irreducible factors of primitive permutation group representation

Consider a primitive permutation group $\Gamma\subseteq\mathrm{Sym}(N)$ on the $n$-element set $N=\{1,...,n\}$, that is, $\Gamma$ does not preserve any non-trivial partition of $N$. Consider the ...
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 $486$ ...
313 views

### Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S_\omega$?

Is $(\mathbb{R},+)$ isomorphic to a subgroup of $S_\omega$, the group of permutations of the set of non-negative integers $\omega$?
253 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 ...
137 views

### Multiplication in $Z(\mathbb{C}S_n)$ [duplicate]

I am trying to multiply two generators of center $Z(\mathbb{C}[S_n])$ of ring algebra of symmetric group of $n$ elements. We know that these generators are given by sums of conjugacy classes in $S_n,$ ...
173 views

### 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 $10$...
69 views

86 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 ...
131 views

309 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 ...
248 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 ...
185 views

156 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 $ax$ ...
101 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....
203 views

452 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). ...
Some definitions and Notations Group Action see this link. $G$-invariant partition $B_i$ means $\forall \sigma \in G, B_i^{\sigma} = B_i$. Minimal block system see page no - 30 Let $V$ be a $k$-...
Let $G$ be a permutation group that acts on (say) $X=\{1,2,..,n\}$, and $H$ be a proper subgroup of $G$. Can one say anything precise about when the number of orbits of $H$ on $X$ will be equal to ...