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.

Filter by
Sorted by
Tagged with
0
votes
1answer
20 views

Can Symmetric group on $n$ letters act non-trivially on a set with $k$ elements where $k<n$?

Let $\Sigma_n$ be the symmetric group on $n$-letters. My question is can $\Sigma_n$ act non-trivially on a set of $k$ elements say $\{1,2,\dots ,k\}$ where $k<n$? I can't think of any examples ...
0
votes
0answers
19 views

Optimal Subgroup for Cayley's Theorem

I want to find the smallest $n$ such that $S_4 × S_8$ is isomorphic to a subgroup of $S_n$. I know that Cayley's theorem will give an $n$ but is there a way in general to find the smallest $n$? If not ...
0
votes
1answer
41 views

Prove $Q_8$ is not contained as a subgroup in the symmetric group $S_n$, for $n<8$

Prove $Q_8$ is not contained as a subgroup in the symmetric group $S_n$, for $n<8$ I think that the key is considering the elements of order $4$ in $S_n$ but Ido not know how to face up to it. In ...
-1
votes
0answers
18 views

The alternating group An does not contain any Sn-1 isomorphic subgroup [duplicate]

It is known that we can define an action of G in the group of permutations of the set of the classes mod H, where H is a subgroup of G. The kernel of that action is the intersection of all H's ...
2
votes
2answers
94 views

Prove there does not exist $\sigma^2=(1, 2, 3, 4)$ for $\sigma \in S_7$

I know how to prove the following using parity ie breaking down the following into its transpositions and then showing it is impossible due to the parity but was wondering if anyone had any ideas on ...
0
votes
0answers
32 views

Image of elements of $S_n$ in the Temperley-Lieb algebra

Consider the algebra $A_n$ generated by $u_1,\ldots,u_{n-1}$ subject to relations $u_i^2=-2u_i$, $u_iu_{j}u_i=u_i$ for $|i-j|=1$ and $u_iu_j=u_ju_i$ for $|i-j| \geqslant 2$. The algebra $A_n$ is the ...
1
vote
0answers
22 views

Constructive aspects of Dilworth's theorem for a class of finite Young's lattices

Dilworth's theorem partitions posets into the so-called chains and states that a poset of width $k$ requires only $k$ disjoint chains to decompose. It is an existential statement but constructive ...
0
votes
2answers
19 views

Stabilizer of diagonal of a cube

The group of rotational symmetries of a cube, $G \cong S_4,$ acts transitively on the set of four diagonals of a cube. Let's denote by $X=\{1,2,3,4\}$ the set of four diagonals. Clearly $|\text{Orb }(...
2
votes
0answers
17 views

Hilbert series for symmetric polynomial ring [closed]

Consider $\mathbb{C}[x_1, x_2, ..., x_n]^{S_n}$. I want to calculate the Hilbert series for this. To do this, I'm trying to find out the dimension of the space of homogeneous symmetric polynomials of ...
1
vote
1answer
58 views

For which $n\geq2$ does there exist $\sigma \in A_n$ with $|\sigma|>n$? [duplicate]

For which $n\geq2$ does there exist $\sigma \in A_n$ with $|\sigma|>n$? I have found this same question here, But I have not been able to understand how it works. On my own I came to the ...
3
votes
2answers
65 views

Computation of irreducible characters for $S_n$ - Mathematica vs. GAP

For some physical applications I need the knowledge of irreducible characters for symmetric groups $S_n$ with large $n$. For small ones I was using FiniteGroupData ...
-2
votes
0answers
12 views

SU(2) symmetry correction problem [closed]

I stumbled on a problem during my study and I don't think I have the ability to solve this myself. I would hope that someone can help me with a solution, or how to get to a solution. I would like to ...
1
vote
1answer
58 views

How to construct an isomorphism $\wedge^2 V \xrightarrow{\sim} \wedge^2 V\otimes U^{\prime}$ as representations of $S_5$?

Let $S_5$ be the symmetric group and $U^{\prime}$ be the $1$-dimensional alternating representation. Let $V$ be the $4$-dimensional standard representation, i.e. $$ V=\{(x_1,x_2,x_3,x_4,x_5)\in \...
1
vote
2answers
33 views

Prime Notation in Transpositions

My textbook has the following problem: Consider the group $S_n.$ Prove that if $\sigma$ and $\tau$ are any pair of distinct transpositions such that $\tau(1)\neq1$ then there exists $\sigma'$,$\tau'$ ...
0
votes
0answers
28 views

Hall Scalar product calculation: $\langle h_3p_6, s_{(5,4)}\rangle$

This is a problem from an old qualifying exam I am reviewing: Calculate: $\langle h_3p_6, s_{(5,4)}\rangle$ where $\langle \cdot,\cdot \rangle$ is the Hall scalar product defined by $\langle s_\lambda,...
1
vote
1answer
50 views

Reflection groups, geometric group theory: John Meier

This image is from John Meier's Introduction to the geometry of infinite groups on p. 47, where reflection groups are being introduced. Now the claim is, that every edge in the tiling can be labelled ...
2
votes
0answers
36 views

the dominance order, its order ideals, and RSK

Let $\Lambda_n$ denote the lattice of all integer partitions of size $n$ where the operative partial order $\lambda \unlhd \mu $ is the dominance order defined by \begin{equation} \lambda_1 + \cdots + ...
1
vote
1answer
21 views

Confusion with number of commuting permutations

Find the number of permutations in $S_6$ which commute with $\sigma = (1 \ 2 \ 3) (4 \ 5)$ This was my attempt: A permutation $\tau \in S_6$ commutes with $\sigma \iff \sigma =\tau^{-1} \sigma \tau =...
1
vote
2answers
70 views

Number of elements of order $p$ in the sylow $p$-subgroup of $S_{p^2}$

I have the group $G = \mathbb{Z}_p \wr \mathbb{Z}_p$, it's well known that $G$ is isomorphic to a sylow $p$-subgroup of $S_{p^2}$, it has order $p^{p+1}$ and there are 2 possible orders for a non-...
2
votes
2answers
36 views

Finding cosets from symmetric subgroup $S_3$ of symmetric group $S_4$ where $S_3 = \{\phi\in S_4\mid\phi(4) = 4\}$

This question confused me a bit. Symmetric group $S_3$ is a subgroup of symmetric group $S_4$ where $S_3 = \{\phi \in S_4 \mid \phi(4) = 4\}$ We were told to find all the different cosets from $S_3$ ...
1
vote
1answer
31 views

A question on Young tableau.

I am reading Fulton's book representation theory. My question occurs in the proof of Lemma 4.23. I will introduce my question concisely without letting you read that book. The book introduces an order:...
0
votes
1answer
52 views

Are the Sylow $p$-subgroups of $S_4$ also Sylow $p$-subgroups of $S_5$?

Here is my thinking: $|S_4| = 4! = 1 \times 2 \times 3 \times 4 = 2^3 \times 3$. $|S_5| = 5! = 1 \times 2 \times 3 \times 4 \times 5 = 2^3 \times 3 \times 5$. Since $2^3$ is the maximal power of $2$ ...
1
vote
1answer
24 views

Is Symmetric group on 5 symbols is the semi-direct product?

Is the Symmetric group on 5 symbols is the semi-direct product of groups $A_5$ and $C_2$, i.e. $$S_5\cong A_5\rtimes C_2?$$ Here $A_5$ is considered as a normal subgroup. Please help.
1
vote
0answers
33 views

Finding that $S_3$ is the homomorphic image of the group with the following presentation

I have $G = \langle x,y,z \mid x^2y^4z^3, x^4y^2, x^2y\rangle $. Now, I have managed to show that its abelianization is isomorphic to $\mathbb{Z}\oplus \mathbb{Z}_3$ but I am really stumped as I am ...
0
votes
3answers
57 views

Find a subgroup of order $120$ in $S_8$ [closed]

Find a subgroup of order $120$ in $S_8$ Listing the possible $k$-cycles in $S_8$. I have that the possible orders of the elements in this group are $1, 2, 3, 4, 5, 6, 7, 8, 10, 12$ and $15$. How can ...
1
vote
0answers
28 views

Distribution of dimensions of $S_n$ irreps for large $n$

A histogram of the dimensions of the irreducible representations of the symmetric group $S_n$ becomes sharply peaked for large $n$. For example, here is a $\log_{10}$ histogram for $S_{50}$: (This is ...
0
votes
1answer
20 views

Unfaithful irreps of $S_n$ with dimension greater than one

The irreducible representation of $S_4$ corresponding to the partition $2+2$ is two-dimensional and unfaithful. Are there other unfaithful irreps of $S_n$ with dimension greater than 1?
4
votes
1answer
45 views

Is there an algebraic characterization of disjoint support in some group action?

Let $\phi:G\hookrightarrow \text{Sym}(X)$ be a faithful $G$-action on a set $X$. Consider $g,h\in G$ s.t. $\text{supp}(\phi(g))\cap\text{supp}(\phi(h))=\varnothing$. It is easy to show that $\phi(g)$ ...
0
votes
0answers
8 views

Direct summand of a group algebra.

I want to show that the matrix ring $M_2(\mathbb{F_7})$ can never be the summand of group algebra $\mathbb{F}_7S_5$, where $S_5$ is the symmetric group on $5$ symbols. To be more precise, I want to ...
0
votes
0answers
13 views

Given two data tuples with seven binary attributes, compute the distance of these two data tuples in Contingency Table

Given the following two data tuples with seven binary attributes, compute the distance of these two data tuples 1) if each attribute is a symmetric attribute, 2) if each attribute is an asymmetric ...
1
vote
1answer
57 views

Sign of permutation who discovered that

I am reading a book about mathematical games. One game is the 15 puzzle https://en.wikipedia.org/wiki/15_puzzle The answer whether it is solvable or not, was solved by using sign of permutations. My ...
0
votes
2answers
56 views

How do I find all homomorphisms between $S_3 \to \mathbb{Z}_{15}$? [duplicate]

How do I find all homomorphisms between $S_3 \to \mathbb{Z}_{15}$? I know that since $S_3 \cong \langle x,y: x^3 = y^2 = e;yx = x^2y\rangle$, the homomorphism is dependent on $\phi{(x)}$ and $\phi{(y)}...
3
votes
0answers
14 views

is this a primitive action of $\operatorname{Sym}(3)$

Let $G=\operatorname{Sym}(3)$ acting on $S=\{1,2,3\}$ and let $S'=\{(x,y): x,y\in S, x\neq y\}$. Now $G$ acts on $S'$ by taking $(x,y)^\rho = (x^\rho, y^\rho)$ for all $\rho \in G$. Is this action ...
1
vote
2answers
72 views

Isomorphism between any finite group and a subgroup of $S_n$

I am trying to prove: Prove that every finite group $G$ is isomorphic to a subgroup of the symmetric group $S_n$ for some $n$. Here is my attempt, which is my attempt at trying to understand a proof ...
1
vote
1answer
56 views

Isometry of cube in $\mathbb R^4$?

Find the Symmetry Group of : Tetrahedron and Cube. I know that there is a duplicate question Symmetry group of Tetrahedron but my professor suggested us to write coordinates of both cube and ...
1
vote
1answer
37 views

How do I prove $\textit{V}_\textit{n}$ is irreducible symmetric group representation?

$\textit{V}_\textit{n}=\left\lbrace (x_1,x_2,...,x_n)\in\mathbb{R}^\textit{n}|\sum_{i=1}^{n}x_i=0\right\rbrace $,we define $ \textit{S}_\textit{n} $ act on $ \textit{V}_\textit{n} $ by $ \textit{S}_\...
2
votes
0answers
41 views

Representations of products of symmetric groups [migrated]

I'm writing a paper and want to cite some references to efficiently prove that over any field $k$ of characteristic zero, every irreducible representation of a product of symmetric groups, say $$ S_{...
0
votes
2answers
40 views

For the symmetric group does a conjugacy class always contain an element and its inverse?

Let $G$ be a finite group and $${\rm Cl}(x) = \{g\,x\,g^{-1}\mid g\in G\}$$ the conjugacy class of $x\in G$. As explained here, in general, the conjugacy class ${\rm Cl}(x)$ does not necessarily ...
-2
votes
1answer
52 views

If $p$ is a prime, then $S_{p-1}$ has no element with order $kp$, $k \in \mathbb{N}$. [closed]

How can I prove the statement: If $p$ is a prime, then $S_{p-1}$ has no element with order $kp$, $k \in \mathbb{N}$. I don't know what is the best approach, I've tried to use permutations, but I can'...
1
vote
1answer
43 views

How to find generator for intersection of two large subgroups (permutations in $S_{13}$)?

I have $$\sigma = ( 1, 2, 3, 4, 5 )( 6, 10 )( 7, 11 )( 8, 12 )( 9, 13 ),$$ $$\tau = ( 2, 5 )( 3, 4 )( 6, 7, 8, 9, 10, 11, 12, 13 )$$ as my two permutations that generate $G$. How would I find the ...
1
vote
1answer
88 views

Prove that for a subgroup $G$ of $S_{13}$, each element can be written as a product of $\sigma^i$ and $\tau^j$.

Given that $G$ is a subgroup of $S_{13}$ (symmetric group) which can be generated with the following two permutations, how can I prove that every element in $G$ can be written as a product of $\sigma$...
2
votes
2answers
56 views

Find the orbit and stabilizer of each element of a set of subgroups

Let $S_3$ be the symmetric group of all permutations on the set $\{1, 2, 3\}$. Then, let $S$ be the set of all subgroups of $S_3$. Consider $S$ as an $S_3$-set with respect to conjugation and for each ...
0
votes
0answers
50 views

$\operatorname{Out}(S_7)=1$ at work.

Let's suppose that $\operatorname{Out}(S_{n\ne 6})=1$ is not a known result. Let's denote with $\mathcal{C}_{(i_1,\dots,i_k)}$ the conjugacy class of $S_7$ corresponding to the cycle type $(i_1,\dots,...
1
vote
1answer
30 views

Definition of transitive group

I need a definition of a transitive group that's accessible to someone who's just started learning group theory (so won't know about actions and orbits etc.). I've written the following: A ...
3
votes
1answer
70 views

Cycle index for $S_2\times S_4$

I am trying to determine the cycle index polynomial of $S_2\times S_4$, for the purpose of finding colourings. This is what I have tried: I computed the polynomials for $S_2$ and $S_4$: $$Z_{S_4}(t_1,\...
1
vote
0answers
53 views

Conjugacy class size of $S_n$

My algebra text writes for the number of cyclic structures (-,-)(-,-) in $S_4$ $\frac{{4\choose 2}\cdot {2 \choose 2}} {2}$ which should be {(1 2)(3 4), (1 3)(2 4), (1 4) (2 3)}. The binomial ...
1
vote
0answers
104 views

Understanding this proof of $\operatorname{Out}(S_{n\ne 6})=1$.

I'd like to have a better understanding of this old valuable answer (I've tried to comment the post, but with no feedback). I'm reading (correctly?): the first two Lemmata as: $\operatorname{Aut}(S_n)...
2
votes
1answer
30 views

What is the order of $U$? Is $\{1, 7\}$ an orbit from $U$?

Let $G: = S_7$ and $U: = \{(17), (1273)\}$. What is the order of $U$? Is $\{1, 7\}$ an Orbit from $U$? Attempt: I know/see $(17)$ has the order $2$ and $(1273)$ the order $4$, but I don't know how to ...
0
votes
2answers
44 views

Show that $G_X\le{\rm Sym}(A)$ where $X\subseteq A$ and $G_X=\{g\in{\rm Sym}(A)\mid g(X) = X\}.$

Show that $G_X \le{\rm Sym}(A)$ where $X \subseteq A$ and $G_X =\{g \in{\rm Sym}(A)\mid g(X) = X\}$ I'm a bit confused with this question and how to approach it. First I believe I need to show $G_X \...
1
vote
1answer
52 views

Understanding proof of Collins' Weingarten calculus formula

I am trying to understand Collins' paper on Weingarten calculus. You can find it here: click! On page 7, between the formulas $(2.6)$ and $(2.7)$, he claims that the action $\pi^{\otimes q} \otimes \...

1
2 3 4 5
46