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Questions tagged [symmetric-groups]

A symmetric group is a group consisting of all permutations of given finite set, with composition of permutations as the binary operation. Should be used with the (group-theory) tag.

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Is there a way to determine whether symmetric group $S_{n}$ has a subgroup of order $m$?

I'm interested in whether $S_{n}$ has a subgroup of order $m$. And as we know, $m | n!$ has to be hold, but it is not a sufficient condition. I have researched some other cases. If $m \le n$, then ...
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Let $H=\{\sigma\in S_n\mid 1\le\sigma(i) \le k\;\forall i=1,2,\cdots,k\}$. Prove that $H$ is a subgroup of $S_n$ and $H\cong S_k\times S_{n-k}$.

I have trouble completing this proof. Let $k,n \in \mathbb{Z}, 0 < k < n.$ Let $H=\{\sigma \in S_n\mid 1 \leq \sigma(i) \leq k\; \forall i =1,2,\cdots,k\}.$ Prove that $H$ is a subgroup of $S_n$...
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General theory for cyclic modules of the group algebra of the symmetric group

Let $\mathbb{C}[\mathfrak{S}_n]$ be the group algebra of the symmetric group. An element of this algebra is of the form $$ v = \displaystyle \sum_{g \in \mathfrak{S}_n} a_g g, $$ where $a_g \in C$. ...
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Conjugate of a Gel'fand pattern

Background: A Gel'fand pattern is a set of numbers $$ \left[\begin{array}{} \lambda_{1,n} & & \lambda_{2,n} & & & \dots & & & \lambda_{n-1,n}...
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Certain subgroups of symmetric groups with trivial intersection [closed]

Let $H$ and $K$ be distinct infinite cyclic subgroups of a symmetric group $S_\Omega$ on a set $\Omega$. Is there an element $g\in S_\Omega$ such that $$(g^{-1}Hg)\cap K=\{1\}.$$ Is it true that for ...
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Prove that $S_4/K \cong S_3$ using the fundamental theorem on homomorphism.

Let $A$ be the set formed by the elements of Klein group but identity, A= { (1,2)(3,4);(1,3)(2,4);(1,4)(2,3)}. Consider the set $Big(A)$ of bijection from A to itseld. With the operation of ...
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5 votes
1 answer
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Number of homomorphisms from $\DeclareMathOperator{\Z}{\mathbb Z}\Z\oplus \Z$ to $S_3$

I read that a homomorphism is fully determined by where its generators are mapped to. Let $f\colon\, \Z\oplus \Z\to S_3$ be a group homomorphism. Generators of $\Z\oplus \Z$ are $(1,0)$ and $(0,1)$. I ...
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Commutativity of the wreath product

Let $G$ be a subgroup of the symmetric group $\mathfrak{S}_n$ and $H$ be a subgroup of $\mathfrak{S}_m$. Recall that the wreath product $G \wr H$ is the semi-direct product $G^m \rtimes H$, where $H$ ...
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Some kind of generalized Leibniz Rule

I want to compute a modified version of the following: Defining $ \partial_i := \frac{\partial}{\partial q_i}$ we can write $$ \partial_{a_1} \cdots \partial_{a_n} \: q_{a_1} \cdots q_{a_n} = \sum_{\...
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What is the full symmetry group of a tile in the shape of a regular n–gon?

I am trying to answer the question on exercise 4.5.6 from the book "Algebra: Abstract and concrete" by Goodman. The chapter is on the symmetries of polyhedra and in this exercise he asks me ...
CoolJedi132's user avatar
2 votes
1 answer
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Basis of the decomposition of two commutating groups

$\newcommand{\ket}[1]{|#1\rangle}$ Consider the following representations of the permutation group $S_n$ and the unitary group $U(d)$ acting on the vector space $(\mathbb{C}^d)^{\otimes n}$ like: $$ P(...
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For $M,C\subset S_n,|M|=|C|$ find subsets of $M$ and $C$ that generate isomorphic subgroups of $S_n$ and the isomorphism maps these subsets together

I have sets $M,C \subset S_n$, s.t. $|M| = |C| \gg 1$. Given $a \in S_n$ I can determine whether $a \in M$ but I have no way to determine whether $a \in C$, however, if we assume C is numbered for ...
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Proving something is a $S_{3}$ submodule

Given that $S_{3}$ is the symmetric group on three elements ${\alpha,\beta,\gamma}$ and that $\mathbb{C}[\alpha,\beta,\gamma]$ is the permutation $G$-module with elements $a\alpha + b\beta + c\gamma$ ...
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Transitivity Theorems - Can You Give Me Some?

Motivation. I get a question, and it reads as follows: "Assume $G$ is an abelian, transitive subgroup of $S_n$" (c.f. Dummit and Foote 4.1.3). Immediately, I can tell you a ton of properties,...
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stab(x) when the Symmetric group acts on the set of order k-tuples

Question: let $G=S_n$ (the symmetric group of degree $n$) and let $X$ be the set of ordered k-tuples $(x_1,…x_k)$ where $x_i \in \{1,2,…,n\}$ (where $n\ge k \ge 1$). G acts on X defined by $\sigma(x_1,...
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How many homomorphisms $C_9 \rightarrow S_4$ are there? [duplicate]

How many homomorphisms $C_9 \rightarrow S_4$ are there? [I want to use the fact that the order of $\phi(g)$ divides the order of $g$ and somehow work from there (I also know that if $\phi$ is ...
Anon314159's user avatar
3 votes
1 answer
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Embedding a Product of Symmetric Groups in a Minimal Symmetric Group

Cayley's Theorem says that any finite group can be embedded in a symmetric group $ S_m $, but it says nothing about the minimum such $m$ for which this is the case. If a finite group is a direct ...
kerbal program's user avatar
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Non trivial subspaces of the permutation module $\mathbb{C}[X]$

Let $X={1,2,...,n}$. Consider the vector space $\mathbb{C}[X]=\{c_1\cdot1+\dots+c_n\cdot n : c_i \in \mathbb{C}\}$ Viewing this as an $S_n$ module, we call this the permutation module. I want to ...
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Omission of 1-length cycles in permutations notation.

I am revisiting some basics on permutations and this might sound trivial but I don't understand what exactly is omitting 1-length cycles in permutations ? Most lecture notes or materials say it's a ...
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The order of the element $\sigma$ of the symmetric group $S_5$ is

The order of the element $\sigma = \begin{pmatrix} 1 &2 &3 &4 &5 \\ \downarrow &\downarrow &\downarrow &\downarrow &\downarrow \\ 4 &5 &1 &3 &...
Hussain-Alqatari's user avatar
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Techniques used to study the structure of groups that are generated by sets that are too big

I have the following set: $M = \{\prod_{k=1}^n {(k,n+y_k)}\in S_{n+m}|(y_1,\dots,y_n) \in \{1,\dots,m\}^n\}$, where $n \geq 2, m \geq 2$ and $(k,n+y_k)$ denote transpositions in $S_{n+m}$. I've shown ...
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The configuration space of 3 unordered points in $\mathbb{R}^2$ with distinct distances

Let $(X, d)$ be a metric space and let $n \in \mathbb{N}$. Define $A_n(X) = \{ (x_i)_{i=1}^n \in X^n \mid \forall i \neq j: x_i \neq x_j \}$ to be the space of $n$ ordered distinct points in $X$. ...
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How to compute the character by "removing the hooks"?

I am reading a paper by McKay in 1971, the name of the paper is Irreducible Representations of Odd Degree. There is a theorem said: ${m}_{2}({S}_{n})=2^r$, where $n=\sum 2^{k_i}$, ${k}_{1}>{k}_{2}&...
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Correctness of algorithm to find the number of elements of order $x$ in Symmetric Group $n$?

To find the number of elements of order $x$ in $S_n$ Generate all possible partitions of $n$ by divisors of $x$. For each partition, check if the LCM of the part lengths matches $x$. Calculate the ...
A. Random's user avatar
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Subset of a conjugacy class of of an odd permutation in $S_n$

Let $\sigma=(1,2,3,\dots,n)$ be an odd $n-$cycle in $S_n$ (so $n$ is even). It is known that the size of its conjugacy class is $|cl_{S_n}(\sigma)|=(n-1)!$. I am interested in the size of the subset $...
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Automorphism group of $A_n$, $n \geq 7$ [duplicate]

I am trying to find the automorphism group of the alternating groups $A_n$. However, when it comes to $A_7$, I have found it difficult to prove that $\operatorname {Aut}(A_7) \cong S_7$. (I have ...
tys's user avatar
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Sums of characters over over partitions of equal length

Let $\chi^{\lambda}$ and $\chi^{\mu}$ be irreducible characters of the symmetric group $S_n$. Their inner product satisfies $\langle \chi^{\lambda}, \chi^{\mu}\rangle =\sum_{\nu} \frac{1}{z_{\nu}} \...
Andrew's user avatar
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Computation of the symmetric number of a finite group [duplicate]

I got a bit curious about the concept of the symmetric number of a finite group, and decided to do some computations with GAP to determine their values for some small finite groups. The symmetric ...
Justin Benfield's user avatar
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1 answer
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A Natural Probability Distribution on the Infinite Symmetric Group

Is there a "natural" probability distribution on the set of bijections from $\mathbb{N}$ to itself? Preferably, I would want a distribution which arises from some combinatorial procedure. ...
Miles Gould's user avatar
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$S_n$ is generated by adjacent transpositions. Are there any other generating sets of length $n-1$ consisting only of transpositions?

$S_n$ is generated by adjacent transpositions. Are there any other generating sets of length $n-1$ consisting only of transpositions? The motivation behind this question is that the set of adjacent ...
H-a-y-K's user avatar
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0 answers
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If GCH is wrong in ZFC...

I am currently working on a presentation about the Symmetric Group of a set. I was able to show that for any infinite set X |Sym(X)|=|Pot(X)| holds (in ZFC). This gives us that in the case that GCH (...
Simon Colt's user avatar
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Using Group action of the $\langle \sigma \rangle$ on $S_p$ for $p$ is prime, show that $S_p$ is cyclically generated by transposition and $p$-cycle

Using Group action of the $\langle \sigma \rangle$ on $S_p$ for $p$ is prime, show that $S_p$ is cyclically generated by transposition and $p$-cycle. Sorry for long title, but this question is widely ...
likely_fail_2202 T_T's user avatar
1 vote
0 answers
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How to randomly construct a long subgroup chain for $S_n$

Description of the problem My task is to build an optimal method that randomly constructs a proper subgroup for a given $G \leq S_n$. Here the term "random" is applied loosely and there is ...
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1 vote
1 answer
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Expected number of connected components if a graph constructed out of two perfect matchings

Let $|V| = n$, an even number of vertices, and let $M$ be a perfect matching on these vertices. Suppose we choose uniformly at random a permutation $\pi$ from the symmetric group $\mathbb{S}_n$, and ...
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How to transform a strong generating set of $G \leq S_n$ to a set that generates a proper subgroup of $G$?

Suppose $G \leq S_n$ and $S$ is a strong generating set of G. Before starting, I'll describe how $S$ is constructed in my particular case. Suppose we have a subgroup chain $G = G_0 \geq G_1 \geq G_2 \...
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3 votes
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Structure of the subgroup of $S_n$ generated by a $n$-cycle and a transposition

Context. This question is a follow-up of the following one. Let $n\geq 1$ be an integer, let $d\mid n$, and let $H=\langle\sigma, \tau \rangle\subset S_n,$ where $ \sigma=(1 \ 2 \cdots n), \tau=( 1 \ ...
GreginGre's user avatar
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6 votes
1 answer
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What is the order of the following subgroup $\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle$ of $S_n$?

Let $n\geq 2$ be an integer, and consider $H=\langle (1 \ 2 \ \cdots \ n), (a \ b)\rangle\subset S_n.$ It is known that $H=S_n$ if and only if $b-a$ and $n$ are coprime. Question. What is the order of ...
GreginGre's user avatar
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4 votes
1 answer
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Symmetry group of the unit circle in $\mathbb{R}^2$ versus $\mathbb{R}/\mathbb{Z}$.

I am studying a set of lecture notes on group theory, and I don't think I understand a point the author makes about the unit circle and its symmetry group in relation to $\mathbb{R}/\mathbb{Z}$. Let $...
JohnT's user avatar
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1 vote
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A subgroup of $S_6$.

Show that the set of permutations $\{\sigma \mid \sigma(i)\le i \text{ for }1\le i\le 6 \}$ is a subgroup of $S_6$. I am stuck at how I am supposed to reason about this question. I'd assume I'd first ...
Jason Xu's user avatar
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1 answer
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Symmetric group representation $S_{3}$ to $\mathbb{C}^{3}$

Trying to find answers for this year french intern agregation (Algebra exam), i found this question about a proof that a particular group representation of $\mathcal{S}_{3}$ on vector space $\mathbb{C}...
Armand Jourdain's user avatar
1 vote
3 answers
94 views

Showing There is No Group Epimorphism from $S_n \longrightarrow \mathbb{Z}/2\times\mathbb{Z}/2$, $n\geq 1$

I know that the function $\text{sgn}: S_n\longrightarrow \mathbb{Z}/2$ is a unique group epimorphism. I am having trouble proving that there does not exist such an epimorphism bewteen $S_n\...
Luk'yan Vilshansky's user avatar
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1 answer
118 views

On the number of $\sigma\in S_{p-1}$ of a given form.

Let $p$ be a prime, $d$ a proper divisor of $p-1$, and $\sigma\in S_{p-1}$ of the form (everything is modulo $p$): $$\sigma=(1,x,\dots,x^{d-1})(i_2,i_2x,\dots,i_2x^{d-1})\dots(i_k,i_kx,\dots,i_kx^{d-1}...
Kan't's user avatar
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1 vote
0 answers
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Number of conjugacy classes in a set of subgroups of $S_5$ which are isomorphic to klein's 4 group.

Let, K be a set of subgroups of S5 (symmetric group of 5 elements) that are isomorphic to the non-cyclic group of order 4. How many conjugacy classes are there in K? I know that a non-cyclic group of ...
Captain_Grothendieck 's user avatar
2 votes
2 answers
72 views

Can we choose a set to make sure the action of a permutation group transitive?

Let a finite group $G$ of order $n$ be given, so $G$ is isomorphic to a permutation group embedded in $S_n$. Can we always find a set $\Omega$ such that $G$ acts transitively on $\Omega$? (For example,...
utx7563yu's user avatar
3 votes
2 answers
163 views

Explicit isomorphism of algebras $\mathbb C[S_3]\cong \mathbb C \times \mathbb C \times M_2(\mathbb C)$

Let $\mathbb C[S_3]$ be the group algebra of $S_3$. We have an isomorphism of $\mathbb C$-algebras $$\mathbb C[S_3]\cong \mathbb C \times \mathbb C \times M_2(\mathbb C).$$ The existence of such an ...
pyridoxal_trigeminus's user avatar
2 votes
0 answers
52 views

An identity about the sum of the reciprocal of the irreducible character of a symmetric group evauated at identity

I just learnt character theory and I am reading the paper https://academic.oup.com/jlms/article-abstract/66/3/623/811347 which is a quite beautiful paper. Section 4.3, on p. 631 contains an inequality ...
Han's user avatar
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0 answers
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Character table of S4

I am trying to understand the character table of $S_4$. I have obtained the trivial, signature and standard representations. The fourth one is the product of signature and standard. Now for the last ...
user519535's user avatar
6 votes
0 answers
96 views

When is the Cayley embedding for infinite groups optimal?

By Cayley's theorem, any group $G$ naturally injects into the symmetric group $\mathrm{Sym}(G)$ of its underlying set via the Cayley embedding. Let's say this embedding is optimal for $G$, if there is ...
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1 vote
0 answers
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Geometric Algebra: No other point groups besides $H_p$ and $C_p$ in two dimensions

I have been reading the paper "Point Groups and Space Groups in Geometric Algebra" by David Hestenes as part of my introduction seminar to GA. On page 5 the remark is made, that to prove ...
user1292126's user avatar
4 votes
1 answer
104 views

Triple-Transitivity/"Specify three know all" property of exotic transitive $S_5\subset S_6$

Let the exotic transitive subgroup $S_5\subset S_6$ act on $\{1,2,\dots,6\}$. For $1\leq i,j\leq 6$, define subsets: $$X_{ji}:=\{\sigma\in S_5\,\mid \sigma(j)=i\}.$$ Does the following properties hold ...
JP McCarthy's user avatar
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