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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Number of involutions in $S_n$? [duplicate]
I was having some fun with the number of involutions (didn't know they were called that) in the symmetric group $\Psi(S_n$), and tried to come up with a simple formula for it, I'm like, so close to ...
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Simple modules for the image of $\mathbb C S_\ell$ in $\mbox{End}(V^{\otimes \ell})$ and Schur-Weyl duality
Let $V = \mathbb C^n$ and let $\{v_1,\cdots,v_n\}$ be a basis over $\mathbb C$. For $\ell\geq 1$, the symmetric group $S_\ell$ acts on the $\ell$-th tensor power $V^{\otimes \ell}$ by permuting the ...
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How does Sage order its symmetric group elements?
In Sage, the symmetric group is a list. For instance if $G$ = SymmetricGroup($3$), we have
\begin{align*}
G[0] & = e \\
G[1] & = (1,3,2)\\
G[2] & = (1,2,3) \\
G[3] &= (2,3)\\
G[4] &...
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Ways to express the identity permutation as precisely $2r$ transpositions in $S_n$
The question is as stated in the title. We know the identitiy permutation can be expressed as the product of even numbers of transpositions, but fixing the number of transpositions and asking how many ...
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Example Calculation: Mackey's theorem for $S_5$
I'm trying to apply Mackey's theorem to a toy example where I start with the trivial $S_3\times S_2$-module $V$, induce to $S_5$ and restrict back to $S_3\times S_2$. The version of Mackey's theorem I'...
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What is multiplicity in isotypic decompositions
I've been looking into the Quantum Schur Transform, first introduced by this paper.
The explanation of the procedure goes like so:
Consider the two groups: $S(N)$, the symmetric group on $N$ elements, ...
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Conjugacy classes in a subgroup of symmetric group
Let $H=\langle (12)(34),(456)\rangle\cong S_4$ be a subgroup of $S_6$. It is known that there are five conjugacy classes in $S_4$ and the representatives are $(1), (12), (12)(34), (123), (1234)$.
I ...
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Must two permutations in $S_n$ that only differ at two adjacent positions have different signs (i.e., one is even and one is odd)?
The Problem: Must two permutations in $S_n$ that only differ at two adjacent positions have different signs (i.e., one is even and one is odd)?
For example, suppose $\sigma, \tau\in S_n$ such that $\...
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Cohomology of symmetric group of order $p!$ with coefficient tensor $p$ times of a graded vector space.
Suppose $\Sigma_p$ is the symmetric group of order $p!$. Let $V$ be a graded $\mathbb{F}_p$-vector space such that $V_i$ is non-trivial for $i$ arbitrary large at even and order degree.
Claim: $H^j(\...
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Intuition on "sharper" Cayley theorem
In "A Book of Abstract Algebra" by Pinter, chapter 16.G is the following exercise:
If H is a subgroup of a group G, let X designate the set of all the left cosets of H in G. For each element ...
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Spanning set of symmetric invariants of tensor powers
Let $M$ be a module over a commutative ring $R$ and let the symmetric group $\Sigma_n$ act on $M^n$ and $M^{\otimes n}$ by permuting factors. For $v \in M^n$, let $\text{Stab}(v)$ denote the ...
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Stabiliser of $(12345)\in S_5$ for conjugation. [duplicate]
Let $\sigma= (12345)\in S_5$. I read that the Stabiliser of $\sigma$ under conjugation is the group generated by $\sigma$.
Can someone explain how to prove this?
I know that $\tau \sigma \tau^{-1}= (\...
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Prove $D_6 \cdot \text{stab}_{S6}(2) \leq S_6.$
I wish to show $D_6 \cdot \text{stab}_{S6}(2) \leq S_6$, where $D_6$ is the dihedral group of order $12$ and $S_6$ is the symmetric group of order $6$.
I have been able to provide the start of a ...
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Numbers of elements of order $p$ in $Sym(n)$
First of all, I saw that question Number of Elements of order $p$ in $S_{p}$ and additionally ask this question. Please inform me if there is such question in the site then we can close question.
...
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What is a $\{2,3\}$-group?
I found the following statement in Cameron's Projective and Polar Spaces (page 120 here):
If two such automorphisms of order $3$ have a common fixed line, then they generate a $\{2,3\}$-group, since ...
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Elements of $A_8$ commuting with $A_5$
How would we go about showing that in $A_8$, an element of order $3$ commuting with a subgroup isomorphic to $A_5$ is necessarily a $3$-cycle?
I know that an element of order $3$ in $A_8$ must be a ...
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$\Sigma\le S_n$ of order $n$. Is $\Sigma$ cyclic, if each $\tau\in\Sigma\setminus\{Id\}$ moves all the elements?
If $\sigma\in S_n$ is an $n$-cycle, then every $\tau\in\Sigma:=\langle\sigma \rangle$ moves all the elements of $\{1,\dots,n\}$. Now I wonder whether the other way around holds, namely:
Claim. Let $\...
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On the cyclicity of $\operatorname{Aut}(\Bbb Z/p\Bbb Z)$, for $p$ a prime.
Let $p$ be a prime. The group $\operatorname{Aut}(\Bbb Z/p\Bbb Z)$ acts regularly on the set of generators of $\Bbb Z/p\Bbb Z$. Therefore, for every $i,j=1,\dots,p-1$, there is one and only one $\...
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Combining Operation for $S_4$ [closed]
If $\sigma=(1\ 2\ 3\ 4)$, $\kappa=(1\ 2)$ for $S_4$ and I want to compute $(\sigma\kappa)^2$, does it become $\sigma^2\kappa^2 = \sigma^2$ (since $\kappa^2 = 1$), which is just $(1\ 3)(2\ 4)$?
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Is the following presentation for $D_8$ a valid presentation?
My professor used the following presentation for $D_8:$
$$D_8 = \{s,t | s^2 = t^2 = (st)^4 = 1\}$$
But I am not sure if this presentation is correct, I looked at subWiki here https://groupprops....
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What's the smallest $n$ for which $S_n$ has an element of order $30?$
This is Gallian 5.22.
According to the manual the answer is $10$, but it's not clear what combination of cycles from $S_{10}$ would give an ${\rm lcm}$ of $30$ (and as such an element of order $30$).
...
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Proof techniques to show a representation is faithful
I am curious what proof method is most commonly used to show that a representation is faithful. I have found remarkably little online about this question..
It makes sense how to show that a ...
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Expected order of an element in $A_5$ [closed]
Suppose we pick every element from the Alternating group $A_5$ equally likely. What would be the expected order of the element?
My answer is $\frac{241}{60}$. This is because, we have $1$ element of ...
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Young diagrams for block matrices
Let $S_n$ be the group of permutations of $n$ elements. Consider the map $S_n \to S_{mn}$ of block permutations, and an irreducible representation of $S_{mn}$ (over the complex numbers), corresponding ...
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Find the most "scattered" permutations
Given $n$ and $k$, we want to find $k$ permutations that are most "scattered".
Background: I am trying to have multiple outputs using a greedy algorithm on a combinatorial optimization ...
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Show that if $A=\{1, 2,...\}$, then $S_A$ is an infinite group
I want to show that if $A=\{1, 2,...\}$, then $S_A$ is an infinite group.
My proposed solution:
We consider the element $x=(1, 2,..., i, i+1,...)$ of $S_A$ , where we have used cyclic notation for ...
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Understanding the Irreps of a Particular Representation of Symmetric Group Product
I'm reading this paper on the element distinctness problem, and I'm having some trouble parsing Claim 2. I've recently been going through The Symmetric Group by Sagan (Chapters 1 and 2 so far).
Here's ...
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Show $S_n/A_n$ is commutative
Let $S_n$ be the symmetric group and $A_n$ the alternating subgroup. I want to show $S_n/A_n$ is commutative.
Given that the index of $A_n$ in $S_n$ is $2,$ the quotient group consists of two elements,...
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Technicality about cycle of a permutation, for coding purposes
A meta-question: If I were to ask, "What is the effect on 3 by the following permutation?"
$\sigma = (1 2)$
Answer one: It sends 3 to 3.
Answer two: It is undefined.
Answer one is correct in ...
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A finite group of order $mn$ with $m,n$ relatively prime, together with subgroups of orders $m, n$.
Let $G$ be a finite group of order $mn$ with $(m,n) = 1$. Assume that there exist subgroups $M,N$ of $G$ of orders $m$ and $n$, respectively. Prove that $G$ is isomorphic to a subgroup of the ...
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Number of fixed points of generators of reflections (Coxeter) group
Say I have a group with presentation like
$$\langle s,t,u \mid s^2,t^2,u^2,(st)^2,(su)^3,(ut)^4\rangle,$$
faithful on set $S$ with exactly one orbit ($|S|$ is known). How could I determine $|\text{Fix}...
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Prove that certain Permutations are even
I'm trying to prove the following statement:
Let $m \geq 3, 2r \leq m+1, \{x_1,...,x_r\},\{y_1,...,y_r\}\subseteq \{1,2,...,m\}, 1\leq s\leq m$ then $\exists \sigma,\rho \in A_m$ with $\sigma x_r = \...
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Combinatorics of a Single-Chiral-Center Molecule
I'll preface with the fact that I'm currently studying undergrad biophysics, so I don't have much background in math.
What I want to find:
a combinatorics approach to calculating the number of unique ...
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Finding the smallest possible $n$ such that $S_{n}$ has an element of a given order. [closed]
I am realizing that I am not sure I quite understand permutations, especially in this context. The specific problem I am working with asks to find the smallest possible $n$ such that $S_{n}$ has an ...
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Unique representation of a graph (graph automorphism) in python
I'm trying to implement a boardgame in python, but I'm having quite a bit of trouble finding a clever way to solve the following graph problem.
(Image to help visualize the game and pieces I'm talking ...
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Let $H = \{g \in S_6 : g(a)\equiv a\pmod3 \text{ for } a\in\{1,2,3,4,5,6\}\}.$ Is this a subgroup?
Let $H = \{g \in S_6 : g(a)\equiv a\pmod3 \text{ for } a\in\{1,2,3,4,5,6\}\}.$ Is this a subgroup?
The identity element $\in H$ because for all $a \in \{1,2,3,4,5,6\}$, we have $h(a) ≡ a\pmod3$ as $h(...
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Subgroup Requirements
Let $H = \{g \in S_5 : g(2) = 2\text{ and }g(4)=4\}$.
Is this a subgroup ? I know the identity element $\in H$ as this is $(1)(2)(3)(4)(5)$ which satisfies the requirements of the original group.
For ...
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Prove $\operatorname{sgn}(\pi) = (-1)^r$ where $r$ is the number of inversions.
EDIT:
PROOF BY INDUCTION: If there are no inversions then we have an identity permutation, so the base case holds.
INDUCTIVE STEP : Lets fix some $r \in \mathbb{Z}^{+}$ and take any $\pi \in S_n$. If $...
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Show $\phi : S_3 \rightarrow S_3$ given by $\phi(f^{i}g^{j}) = f^{2i}g^{j}$ is an automorphism
Show $\phi : S_3 \rightarrow S_3$ given by $\phi(f^{i}g^{j}) = f^{2i}g^{j}$ is an automorphism
I'm not really sure how to begin.
I know that I need to show that $\phi$ is a group isomorphism, but the ...
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transitive subgroups in $S_n$ that are isomorphic to $S_k$ for $k\leq n$
Assume that there is a transitive subgroup $H$ of $S_n$ w.r.t. the standard action $S_n\curvearrowright \{1,2,\cdots, n\}:=X$ such that $H\cong S_k$. Is there any sharp estimate on the upper bound of ...
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Is there any nonabelian group of order $n$ that isn't a subgroup of $S_{n-1}$? [duplicate]
As far as I know, the only groups of order $n$ that aren't subgroups of $S_{n-1}$ are cyclic groups with prime power order and the Klein four-group. Is there any nonabelian group of order $n$ that isn'...
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Product of Permutation Representation Characters
Consider the action of $S_n$ on $x_i$, where $x_i$ is a set of $i$-element subsets of $X =$ {$ 1, 2, ..., n$} (so $|x_i| = {n \choose i}$).
Now, let $\pi_i$ be the character of the permutation ...
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Extending the hamming distance on Sym$(X)$ to linear groups
I was reading of the hamming distance on the symmetry group of a finite set $X$. The distance was defined as follows: $$\text{d}^H (\sigma_1 , \sigma_2) = \frac{| { x \in X : \sigma_1(x) \neq \sigma_2(...
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Symmetric group characters
Is there any relation between Littlewood–Richardson coefficients and the characters of the symmetric group? Can there be a relation between $\chi_\lambda(\mu)$ and $c^\lambda_{\mu \mu}$.
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Symmetric vector least squares solution
I have the similar problem as the Symmetric linear least squares solution. The least square problem of mine is that I want to find
$$
minimize || Ax-b ||^2,
$$
$$
where A\in m\times n,
$$
$$
x,b \...
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Linear involution for Specht modules
Let $n$ be a positive integer and $\lambda$ be a partition of $n$, which we identify with its Young diagram. Let $S^{\lambda}$ be the Specht module associated to $\lambda$.
Here the Specht modules are ...
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Show that for even integers $ n $ there exists an element $ g \in D_{n} $ : $ \operatorname{ord}(g)=2 $ and $ \operatorname{sgn}(\varphi(g))=1 $
Let $ n>3 $, let $ \Delta_{n} $ be a regular $n$-corner, and let $ D_{n} $ be the dihedral group which is the symmetry group of $ \Delta_{n} $. If we number the vertices of $ \Delta_{n} $, we ...
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Find the number of elements of $A_4$ composed of the product of two non-disjoint transpositions.
Find the number of elements of $A_4$ composed of the product of two non-disjoint transpositions.
There are following $12$ elements in $A_4=\{e, (123), (132), (124), (142), (134), (143), (234), (243), ...
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Issue with definitions for conjugacy classes.
I'm trying to understand a proof about the splitting criterion for the conjugacy classes of $A_n$, but I think I'm getting bogged down in earlier terminology. Here's a setup.
Definitions: Let $G$ be ...
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Verify proof: Prove that the order of an element in $S_n$ equals the least common multiple of the lengths of the cycles in its cycle decomposition.
I'm practicing for my upcoming Abstract Algebra midterm, and am trying an exercise in the Dummit and Foote Abstract Algebra book. I would like to verify my solution to the following exercise in ...
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