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.
22 questions from the last 365 days
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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
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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 $p_i$ ...
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The tilde species
Consider a combinatorial species $F$, that is, an action of the symmetric group $\mathfrak S_n$ on a finite set $F[n]$. Recall that the elements of $F[n]$ are called structures. Furthermore, recall ...
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An algorithm to decompose a directly indecomposable permutation group into a wreath product
I am considering the following two binary operations on permutation groups:
the direct product, and
the wreath product.
It turns out that there is an efficient algorithm to factor a given ...
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What do we know about the action of the symmetric group by conjugation on the set of permutation groups?
Motivation:
I have co-authored a package for sagemath to compute with combinatorial species, also known as sequences of group actions of the symmetric groups. In an effort to find good tests for that ...
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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 ...
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Properties of group action of normal subgroups
I'm currently reading Dixon's "Permutation groups". It states that when a group $G$ acts transitively on a set $X$ (let's denote this as $G\curvearrowright X$), the orbits of a normal ...
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Does the Okounkov-Vershik approach to the representation theory of $S_n$ shed new light on the problem of computing Kronecker coefficients?
I am studying the problem of decomposing tensor products of irreps of $S_n$. As a non-expert, I was surprised to see that this is an open problem, or a the very least, one for which no satisfactory ...
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Generating set of permutation group such that generators do not "contain" other group elements
Let $(G, X)$ be a permutation group with domain $X$. Let $O=\{o_1,\dots,o_m\}$ be the set of orbits of $G$. I am interested in generating sets $S$ with the following property:
Let $g\in S$ be a ...
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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 ...
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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
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Is this known? As $p,q\to\infty$, most elements of the power set of $\{1,\dots,p\}\times\{1,\dots,q\}$ are in free $\Sigma_p\times\Sigma_q$-orbits
Let $p,q$ be nonnegative integers. The product of symmetric groups $\Sigma_p\times\Sigma_q$ acts on the power set $P(\{1, \dots ,p\}\times\{1, \dots ,q\})$ in the evident way. You can ask what ...
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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
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What are the possible symmetry groups of n-point constructions in the projective plane?
Let $k$ be an infinite field, perhaps take $k = \mathbb{C}$ if it simplifies matters.
I will be asking a question about $\mathbb{P}^2$ for definiteness and to simplify definitions/notations, but feel ...
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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 ...
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What can be said about these tensor representations of $\mathrm{SL}(2)$?
Let $W = V \otimes \dots \otimes V$, the product of $n$ copies of $V = \mathbb{C}^2$. Let $G$ and $H$ be two subgroups of the symmetric group $S_n$ and let $\chi$ be a character of $G$.
Associated to $...
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Does the permutohedron satisfy any minimal distortion property for graph metric vs Euclidean distance?
We can look on the permutohedron as a kind of "embedding" of the Cayley graph of $S_n$ to the Euclidean space. (That Cayley graph is constructed by the standard generators, i.e. ...
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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 ...
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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 $$\...
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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 ...
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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 ...
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7
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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.
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