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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4answers
45 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
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0answers
41 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
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1answer
42 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. ...
2
votes
5answers
59 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 = ...
2
votes
1answer
49 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
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0answers
33 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?
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0answers
33 views

If $G$ is a subgroup of $S_n$ and $ G \not\subseteq A_n$, is it true that $ G A_n=S_n$ [closed]

If $G$ is a subgroup of $S_n$ and $ G \not\subseteq A_n$, then is it always true that $ G A_n=S_n$.
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0answers
12 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
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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
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1answer
20 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
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1answer
24 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
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0answers
20 views

$Aut(S_n)=Inn(S_n)$ for all natural number n except 6. [duplicate]

Since $G/Z(G)$ is isomorphic to $Inn(G)$ and $Z(Sn)=1$, If $Aut(S_n)=Inn(S_n)$, then $S_n=Aut(S_n)$. How can I show that $Aut(S_n)=Inn(S_n)$ for except $n=6$?
1
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1answer
20 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
76 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 ...
0
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0answers
23 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
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1answer
15 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
46 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 $G$ ...
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0answers
24 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
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1answer
23 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
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1answer
24 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 ...
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0answers
37 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
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1answer
32 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
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3answers
68 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 ...
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0answers
82 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
28 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
58 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 $\mathbb R^n$ ...
9
votes
0answers
79 views

Restricting irreps of $S_n$ to $D_n$ of order $2 n$

I would like to know how to restrict the irreps of the symmetric group $S_n$ to the dihedral group $D_n$ of order $2 n$. We know that $D_n < S_n$. Symmetric group $S_n$ Due to Hardy and Ramanujan ...
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1answer
32 views

Why is Frobenius norm related to the inner product of characters?

This is a continuation of my question asked here. I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the ...
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1answer
19 views

Probability of measuring the label of representation in quantum Fourier transformaton

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In Definition 2, the authors start with the following function. $$ f : G \to \mathbb{C} $$ Then ...
1
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1answer
30 views

Why is the sum of irreducible representations nonzero only when the irreducible representation is trivial?

I am going through Normal Subgroup Reconstruction and Quantum Computation Using Group Representations. In section 3, the authors discuss the probability of measuring the irreducible representation ...
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0answers
60 views

When can a dihedral group $D_{n}$ of order $2 n$ be a $p$-group?

A $p$-group is a group where the order of every group element is a power of the prime $p$. The presentation of a dihedral group $D_n$ of order $2 n$ is as follows. $$D_n = \langle x, y \mid x^n = y^2 ...
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votes
1answer
31 views

Find commuting elements within a permutation group

The question is like this: IF $G=S_5$ and $g=(1\quad 2\quad 3)$, determine the number of elements in $H=\{x\in G:xg=gx\}$. To do the question, first it says $$x(4)=(x(1\quad 2\quad 3))(4)=(1\quad ...
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1answer
36 views

Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
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0answers
22 views

Cycle structure of the generators of the dihedral group

Would the following be correct about generating the dihedral group $D_n$ by permutations? If $n$ is even, the group can be generated as $\langle(2\quad n)(3 \quad n-1) \ldots (\frac{n}{2}-1 \quad ...
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1answer
27 views

characters in semi-direct product.

The character tables of the irreducible representations of $T_d$ and $C_{3v}$ are linked. In the notation on those pages, $A_1$ and $A_2$ are irreducible representations of degree 1, $E$ is degree 2 ...
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1answer
17 views

Why doesn't the coset (1,4,2,3)K belong to the Quotient group

I had been given the following question and answer ( in the image) However i do not understand, why for example: (1,4,2,3)K does not belong the the quotient group? Is there any faster way of ...
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2answers
50 views

Counting the number of “distinct” permutations of two sets?

I don't really know how to introduce this question, so I start defining something I needed in order to well understand the problem I met! Let $A$, $B$ two finite sets of distinct elements, with ...
0
votes
1answer
35 views

Group action and equivalence relation

Let $G$ be finite, and group action on $X\subseteq G$: $g\cdot x:=g^{-1}xg$. Let $G=S_n$, and $X=S_n.$ Show that $[x]_R$ consists of all elements of $S_n$ that are of the same cycle-type as $x$. I ...
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0answers
18 views

Vertex-transitivity of the automorphism group of a digraph

I am trying to understand the theorem 3 of Cycles in graphs and groups by Kantor. Theorem $3$ If $G$ is a vertex-transitive group of automorphisms of a digraph $\Gamma$ with outdegree $d \ge 1$, ...
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0answers
55 views

Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...
0
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1answer
13 views

Count all the function f satisfying followings : [duplicate]

For $A=\{1,2,3,4,5,6,7,8,9,10\}$ Define a function $f : A\to A.$ Then 30 times composite of f, that is ; $f\circ f\circ...\circ f(x) = x$ and 30 is the least number for f to become an identity. How ...
0
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1answer
51 views

How many number of functions are there?

$A=\{1,2,\dots,10\}$ Define $f:A \rightarrow A$ then $f^{30}(x) = x$ ($30$ times composite of $f(x) = x$ and the number $30$ is the least number for $f$ to become an identity function) How many ...
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1answer
33 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
3
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1answer
45 views

What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
0
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3answers
30 views

Normality is not transitive

Let $G=S_3\times S_3$ where $S_3$ is the symmetric group. Let $p= \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ \end{pmatrix} $, let $L=(p)$, $K=L\times ...
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0answers
13 views

Finding the character table for Z_8

I am a bit confused about how to come up with the number of irreducible representations, as well to come up with the number of different conjugate classes. Starting me out would be highly appreciated ...
2
votes
1answer
42 views

Number of Elements in a Conjugacy Class of $S_N$ (Derivation)

Consider the conjugacy classes of the symmetric group $S_N$. Each conjugacy class consists of permutations that have the same cycle structure. We see that the number of possible cycle structures is ...
3
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0answers
25 views

Fixed-point subspace of $O(2)^-$, a subgroup of $O(3)$

$O(2)^-$ is generated by the $SO(2)$ of rotations about the $z$-axis and a reflection through a vertical plane. The space $V_l$ is generated by spherial harmonics, i.e., Cartan decomposition ...
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vote
2answers
30 views

Decomposition of permutation

I was asked to decompose the permutation $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1 \\ \end{pmatrix} = (12345) \in S_5$$ into a product of two ...
2
votes
0answers
19 views

Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...