# 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.

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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 ...
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### 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 ...
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### 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 $W = V\wedge V$ (...
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### 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 ...
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### 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,...
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### 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,...
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### 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 $\mathbb{P}^1(k)$, ...
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### 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 ...
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### 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 ...
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### 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 $486$ ...
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### 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 $\omega$?
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### 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 ...
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### 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 $S_n,$ ...
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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 $10$...
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Consider a 3*3 matrix $\left( \begin{array}{*{20}{c}} {{m_{11}}}&{{m_{12}}}&{{m_{13}}}\\ {{m_{21}}}&{{m_{22}}}&{{m_{23}}}\\ {{m_{31}}}&{{m_{32}}}&{{m_{33}}} \end{array} \right)... 4 votes 1 answer 130 views ### Diameter for permutations of bounded support Let$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$be the graph whose ... 4 votes 1 answer 129 views ### Example of primitive permutation group with a regular suborbit and a non-faithful suborbit I would like some examples of groups$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$is ... 28 votes 2 answers 1k views ### Does the symmetric group$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$. In particular, note that neither subgroup is required to be normal, thus making this a ... 9 votes 0 answers 142 views ### Permutation groups with diameter$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$... 11 votes 1 answer 261 views ### Permutation groups having a regular cyclic subgroup and a conjectured algebra of characters Let$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$and let ... 5 votes 1 answer 444 views ### Number of permutations that are products of disjoint cycles of distinct length What is the number of permutations$\pi\in S_n$that are products of disjoint cycles of distinct length? What is the number of permutations that are products of disjoint cycles such that no more than$...
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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 ...