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

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3
votes
2answers
29 views

How many elements in $S_{8}$ are conjugate with $(12)(345)$?

How many elements in $S_{8}$ are conjugate with $(12)(345)$? My reasoning is as follows: Two elements in $S_n$ are conjugate if and only if they have the same cycle type, so we need to count the ...
0
votes
1answer
28 views

Sylow subgroup of a symmetric group

Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which ...
-2
votes
0answers
36 views

Simmetric group exercise of an exam [on hold]

In $S_5$ let be $\sigma=(12435)$ and $\tau=(25)(34)$, and $H= <\sigma, \tau>$. Show that $N(<\sigma>) = H$ N.B. $N(<\sigma>)$ is normalizer in $H$, not in S5. Deduce that $H=&...
1
vote
0answers
31 views

The relation between Weyl character formula and Frobenius characteristic map

Let $\mathfrak{gl}(n)$ be the general linear Lie algebra of rank $n$, and $\mathfrak{S}_d$ be the symmetric group of rank $d$. It is well-known that the Schur-Weyl duality provide a equivalence ...
1
vote
1answer
30 views
+100

Citations for the proof of universality of graph classes

In Automorphisms of graphs, Peter J. Cameron mentioned following classes of graphs which are universal structures. graphs of valency k for any fixed k > 2; bipartite graphs; strongly regular graphs; ...
0
votes
1answer
32 views

Finding elements in $S_{3}$

Question: In $S_{3}$, find elements $\alpha$ and $\beta$ such that $\left | \alpha \right |=2,\left | \beta \right |=2$ and $\left | \alpha \beta \right |=3$ I note that the permutation in $S_{3}$ ...
4
votes
1answer
39 views

Does the commutator group of $S_n$ equal $A_n$ in general?

And how would one deduce this? $[S_n, S_n]$ consists of even permutations so it's obvious that $[S_n, S_n] \leq A_n$, but is $[S_n, S_n] = A_n$ true as well? If so, how to deduce this? If not, how ...
2
votes
1answer
69 views

If $G=\left<(12),(34),(45)\right>\subset S_5$, then $G\cong C_2\times S_3$

Let $G=\left<(12),(34),(45)\right>\subset S_5$. Show that $G\cong C_2\times S_3$. So my first idea was to set $a=(12)$, $b=(34)$ and $c=(45)$ and remark that $$G=\left<a,b,c\mid ab=ba,ac=ca, ...
1
vote
1answer
39 views

Irreps of products between dihedral group and any finite group

Let $D_n$ be the dihedral group with order $2 n$. The total number of irreducible representations for $D_n$ is as follows. When $n$ is even, the total number is $\frac{n-2}{2} + 4 = \frac{n}{2} + 3$. ...
0
votes
1answer
54 views

Non-trivial graph automorphism groups with $D_n$ as subgroup

I understand that the automorphism group of an $n$ cycle graph is the dihedral group $D_n$ of order $2 n$. From the comment of @Christian, I also understand that $S_n$ is the automorphism group of the ...
0
votes
1answer
35 views

Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
0
votes
0answers
10 views

Inner product of Induced permutation representation and an irrep $\langle {\chi \uparrow^{S_n}_{D_n}}_{\mathbf{ 1}_{D_n}} , \chi_\rho \rangle_{S_n}$

I am trying to compute the inner product of the characters of the induced permutation representation from the trivial representation of a dihedral group $D_n$ of order $2 n$ to $S_n$ and an irrep $\...
0
votes
1answer
33 views

A subgroup of $S_n$ having index 2 is $A_n$. [duplicate]

I want show that If $ n \geq 3$ , A subgroup $H$ of $S_n$ having index 2 is $A_n$. Say $H$ is such a subgroup. Since $A_n$ is a normal subgroup of $S_n$,$H$ normalizes $A_n$ . By 2nd ...
0
votes
1answer
31 views

Which vertex-transitive planar graphs represent non-self-intersecting polyhedra?

Consider an infinite planar graph with the following properties. Its vertices all have valence $3$. The faces all have $5$ edges. Now put it in cartesian space and require that the faces are all ...
0
votes
0answers
26 views

Permutation representation for low degree

Thanks for any answer. Suppose $n\leq 10$ and $n\neq 6$ and $k\geq 3$. How can I find all faithful permutation representation of $S_k$ in $S_n$? I mean is there any faithful representation except ...
2
votes
0answers
49 views

Show that $\mathcal D_{12}\cong C_2\times \mathfrak S_3$ [duplicate]

Let $\mathcal D_{12}=\left<a,b\mid a^6=b^2=1,b^{-1}ab=a^{-1}\right>$, $\mathfrak S_3=\left<(12),(123)\right>$ and $C_2=\left<g\mid g^2=1\right>$. I would like to show that $\mathcal ...
1
vote
0answers
30 views

Is there a simple way to find the conjugacy classes of $A_n$? [duplicate]

For the symmetric groups $S_n$ there's a quite simple way to find all the conjugacy classes. We find partitions of $n$, that is, we write $$n = n_1+\cdots+n_k,$$ and then for each such partition ...
2
votes
2answers
95 views

Counting Circular Sequence (Burnside Lemma?)

How many distinct circular binary sequences of length $n$ are there? How many distinct circular binary sequences of length $n$ containing a given pattern, e.g., $110$ are there? The same questions as ...
2
votes
0answers
21 views

Symmetrized monomials under Weyl group?

Consider a given partition $\lambda=(\lambda_1,\lambda_2,...,\lambda_N)$ and start with the monomial $$z_1^{\lambda_1}z_2^{\lambda_2}...z_N^{\lambda_N}$$ in $N$ variables $z_1,z_2,...,z_N$. Now we ...
1
vote
0answers
28 views

Showing $PSL(2,9)$ doesn't have a subgroup of order $90$

I got stuck in proving that there is not any subgroup of order $90$ in group $PSL(2,9) = SL(2,9)/Z(SL(2,9))$. Can anyone help?
0
votes
0answers
21 views

How can I prove that $PSL(2,3)\simeq A_4$?

I would like to prove that $PSL(2,3) \simeq A_4$. I know the structure of $SL(2,3)$ and that $A_4=V\rtimes C_3$, where $V$ is the Klein subgroup. How can I proceed? Thanks for the help!
0
votes
2answers
28 views

Transitive group not necessarily have an $n$-cycle

I know that it is true that if $G$ has an $n$-cycle, then $G$ is transitive as a subgroup of $S_n$, but now I'm trying to find an example why the converse is false. I've been trying examples with $...
0
votes
0answers
21 views

Difference between symmetry algebra and symmetry group

What is the difference between symmetry algebra and a symmetry group? I just wanted to know if my understanding is right. Lets say we have a system of differential equations. Then the symmetry group ...
1
vote
2answers
57 views

If $A$ is a matrix with negative eigenvalues, then $\exists M$ : $A = -MM^T$

Let $A$ be a symmetric matrix with all its eigenvalues negative. Prove that there exists a matrix $M$ such that : $A = -MM^T$. Now, regarding my question, I have found another older question, that ...
3
votes
2answers
97 views

$(C_2 )^3$ is not a subgroup of $S_4$

Prove $(C_2)^3$ is not a subgroup of $S_4$. (Using group actions.) I could think of a permutation argument that $(C_2)^3$ is not a subgroup of $S_4$. But I would like to argue it by considering ...
3
votes
2answers
27 views

Show that the product of two transpositions can be expressed as a product of $3$-cycles

Consider the symmetric group $S_n$ where $n>2.$ Show that the product of two transpositions $(ab),\,(cd)$ can be written as a product of $3$-cycles where $a,b,c,d$ are all distinct. I'm not sure ...
2
votes
4answers
54 views

Permutations conjugated

Show that the permutations: $\alpha= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 5 & 3 & 6 & 1 & 4 \\ \end{pmatrix} $ and $\beta= \begin{pmatrix} 1 &...
0
votes
0answers
46 views

Computing the characters of $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$

How can I compute the characters of the induced representation $\mathbf{1}_{D_n} \uparrow^{S_n}_{D_n}$? Here, $S_n$ is the symmetric group over $n$ symbols and $D_n$ is the dihedral group of order $2 ...
0
votes
1answer
48 views

The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. Since ...
2
votes
5answers
64 views

Proving that the groups $S_3$ and $D_6$ are isomorphic [duplicate]

In particular, $S_3$ is the group of permutations of $\{1,2,3\}$, and $D_6$ is the dihedral group of symmetries of the triangle (written as $D_{2\cdot 3}$). In generator-relation form, $D_6 = \left<...
2
votes
1answer
51 views

Characters of permutation representations for $S_4$

I am going through the lecture note How to get character tables of symmetric groups. On page 2, it computes the character table of $S_4$. The procedure starts with building the table of the ...
0
votes
0answers
36 views

Simple subgroups of a symmetric group

In class we used in an exercise that "the only simple subgroup of a symmetric group (if it has one) is the alternating subgroup". But I don't understand where this comes from. Can someone help me?
0
votes
0answers
15 views

Considering $Res^G_{H_\rho}$ instead of $G$ in quantum Fourier sampling

I am going through the proof of theorem 4 in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. Here, they are trying to calculate the probability of measuring the ...
1
vote
1answer
16 views

How does a symmetric group act on a set?

Let $S_3$ act on the set $A = \{ (i,j) \ |\ 1 \leq i,j \leq 3 \}$ by $\sigma \cdot (i,j) = (\sigma(i), \sigma(j))$. Find the orbits of $S_3$ on $A$. So I know I have to find a set $\{\sigma \cdot (...
3
votes
1answer
21 views

Show that a set is a right transversal for $S_3$ in $S_4$

I am given the set A ={e,(14),(24),(34)} I'm supposed to show that this set is a right transversal for S3 in S4, meaning that every right coset of S3 contains exactly one element of A. I'm getting ...
0
votes
1answer
26 views

Show that if $K$ is a normal subgroup of $H$ such that $H/K$ is abelian, then K contains all of the 3-cycles.

Claim: For $n≥5$, if $H$ is a subgroup of $S_n$ which contains all of the 3-cycles, and $K$ is a normal subgroup of $H$ such that $H/K$ is abelian, then K also contains all of the 3-cycles. Attempt: ...
1
vote
1answer
21 views

Symmetric group acting on polynomial

This is what's in my book: Let $\Delta$ be a polynomial given by: $\Delta = \prod_{1 \leq i \leq j \leq n} (x_i - x_j)$ For example, when $n=4$: $\Delta = (x_1 - x_2)(x_1 - x_3)(x_1 - x_4)(x_2 - ...
2
votes
3answers
78 views

Show that $S_5$ does not have a quotient group isomorphic to $S_4$

Show that $S_5$ does not have a quotient group isomorphic to $S_4$. If we to assume that $H$ is such a group, than $H$ must be normal in $S_5$ and $|H|=|S_5|/|S_4|=5$. So $H$ must be isomorphic to $\...
0
votes
0answers
24 views

Dimension of induced representation in $S_3$

Let $G=S_3$. It has 3 irreducible representations: $1, sgn$ and $V$; the trivial rep, sign rep and rep $V$ where $dimV=2$ Consider the subgroup $H=S_2$ with irreps $1_H$ and $sgn_H$ What is the ...
-1
votes
1answer
18 views

The order of the normalizer of a $p$-subgroup of $S_{p}$ [closed]

I found it In Exercise in abstract algebra by Dummit and Foote. Let $P$ be a Sylow $p$-group of $S_p$. What is the order of $N_{S_p}(P)$?
2
votes
1answer
52 views

About transitive subgroups of symmetric group $S_n$

When I am studying Galois theory I came across some problems: Let $S_n $ be the symmetric group on $n$ letters($|S_n|=n!$).How to determine all the transitive group $G$ of $S_n $ ( A subgroup $G$ ...
1
vote
0answers
25 views

Ordering of elements in the base of a group

In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset. That ordering, $\prec$, was defined in section 4.6 as follows. Throughout ...
0
votes
1answer
27 views

Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...
1
vote
1answer
30 views

Permutation and Linear Representation of Finite Group

By a permutation representation of a finite group $G$, we mean a homomorphism from $G$ to $S_n$, the (full) permutation group on $n$ letters. By a linear representation of a finite group $G$, we mean ...
1
vote
0answers
38 views

How to determine whether or not $S_{n}$ contains a subgroup of order $m$

I'm currently going through group theory practice problems and I needed some assistance with the following exercise: Does $S_{9}$ have a subgroup of order $25$? Is it correct to just use Lagrange's ...
1
vote
1answer
33 views

Order of subgroup of symmetric group

Let $X$ be a finite set, i.e. that $|X| = n$, and let $G = \operatorname{Sym}(X)$ be the symmetric group on $X$. Let $Y \subseteq X$ be a subset of $X$ and define the subset $G_Y \subseteq G$ to be ...
4
votes
3answers
89 views

Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
2
votes
0answers
85 views

Dense and turbulent orbits

In their 2006 paper "Turbulence, amalgamation, and generic automorphisms of homogeneous structures" Kechris and Rosendal (see here for the arXiv version of the paper) state the following proposition ...
4
votes
1answer
29 views

Counting permutations in $S_n$ with $1,2,..,k$ all in same cycle

The number of permutations in $S_n$ for which the first $k$ items $1,2,...,k$ are all in the same cycle can be shown (by a somewhat tedious argument) to be $n!/k.$ I'm looking for less computational ...
5
votes
1answer
61 views

A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n $

(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...