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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About the notation of composition of permutations in Lang's book

In Lang's "Algebra", p.30-31, I'm confused about the order of reading the composition of two permutations. In p.30, it seems that we read it from left to right (see the bottom equations), but for p.31,...
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10 views

Levi civita symbol identity with n dimension

There is an identity $\displaystyle{\epsilon_{i_1...i_k i_{k+1}...i_n}\epsilon_{i_1...i_kj_{k+1}...j_n} =k!\epsilon_{i_{k+1}...i_n }}$ in wikipedia. https://en.wikipedia.org/wiki/Levi-Civita_symbol ...
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1answer
14 views

If $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$

Question In my group theory course, I am asked to show for $\sigma \in S_n$ that if $\sigma (1 \cdots n) = (1 \cdots n) \sigma$ then $\sigma = (1 \cdots n)^i$ for certain $i$. My answer Let $\sigma \...
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1answer
20 views

Are all sets of $n$, s.t. $R(m)^n=I$, where $R(m)$ is any sequence of $m$ moves on a Rubik's cube and $I$ is the identity operator, known?

I've written a program that finds the number of times, $n$, one must apply any operation $R_i(m)$, which consists of $m$ single moves/turns/elementary operations on a Rubik's cube, s.t. $R_i(m)^n=I$, ...
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1answer
15 views

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite?

Will $n$ for $A^n=\mathbb{I}$, where $A$ is any finite operation on a finite group and $\mathbb{I}$ is the identity operator, always be finite? Consider for instance a finite sequence of moves (...
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Every permutation is a product of two permutations of order 2

I am trying to solve a problem, not for homework, and it has me stomped! For $n\geq 4$ and $\alpha\in S_n$, $$\alpha=\dot{\alpha}\dot{\beta}$$ where $\dot{\alpha},\dot{\beta}$ are of order 2. I know ...
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1answer
27 views

Homomorphism between S4 and A4

I'm asked to find a group G with a subgroup H such that there is no normal subgroup N of G which performs: G/N =~ H. I thought of G=S4 and H=A4, because I don't think there is an homomorphism from S4 ...
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1answer
93 views

Why ${S_{n-1}}$ is not a subgroup of ${S_n}$

I know that in general this is not true. Also, I know that subgroups of index n of ${S_n}$ are isomorphic to ${S_{n-1}}$ but why they are not subgroups? For example, why ${S_6}$ is not a subgroup of ${...
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Sylow $p$-subgroups in $S_p$ and order $p$ elements

I realise this question has been asked a few times before, but I don't understand the answers. What is the number of Sylow p subgroups in S_p? Why are the order $p$ elements in $S_p$ exactly cycles ...
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2answers
66 views

Books on Group Theory.

I am looking for a book/note that has a good collection of advance results,theorems on finite group theory. By advance, I mean theorems after Sylow,Lagranges(I am considering theorems of Sylow,...
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49 views
+250

Upper-bounding a sum over non-identity permutations

EDIT: Question 1 has been settled (below). The bounty is for question 2. Let $n\geq 3$ and consider the following function $f:S_n\backslash\{e\}\rightarrow \mathbb{R}$ $$f(\sigma)=\sum_{i=1}^n\frac{...
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1answer
51 views

Understand a part of the proof about permutations in a symmetric group on $n$ elements

Let $\sigma$ be an even permutation in $S_n$($\sigma \in A_n$). Assume $\sigma = \tau\sigma\tau^{-1}$ for some $\tau \in S_n$ and assume that the type of $\sigma$ consists of distinct odd integers. ...
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1answer
28 views

Permutation Product

Here is a problem from Algebra by Michael Artin: $p = \left( {\begin{array}{*{20}{c}} 3&4&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&5 \end{array}} \right),q = \left( {\...
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0answers
64 views

Growth of the characters of finite permutation groups in the number of symbols

I have the following questions. When can the characters of the irreducible representations of the elements of a finite permutation group increase exponentially in the number of the symbols the ...
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2answers
48 views

Set acting like a Group

I am a little confused with familiar things, so I am looking for some help. Description: Consider the set, $S_3= \{(123), (132), (213), (231), (321), (312)\}$, a symmetric group acting on $3$ ...
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44 views

Prove /Disprove: Existence of Symmetric Subgroup of Order $n^2-1$ of $S_n$.

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where order of $G$ is $n^2-1$. Problem: Prove (or disporve), Such $G$ exists for $S_n$ for finite $n$. For $n=5,11,71$, ...
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3answers
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Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$

$G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where $n=5, 11, 71$ and order of $G$ is $n^2-1$. Question: Does $G$ (as defined above for $n=5, 11, 71$) exist? How can I ...
2
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1answer
33 views

Permutation of Disjoint Sets of a Symmetric Group

Problem Description: Consider a symmetric group $S_n$ acting on $n$ objects. We partition $S_n$ into two sets $A, B$ such that $A \cap B= \emptyset$ and $A \cup B = S_n$. In other words, $S_n$ is ...
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1answer
42 views

Existence of surjective map / homomorphism from an infinite group onto its symmetric group

From Cayley's theorem , we know that any group $G$ can be embedded in its permutation group $S(G)$ ; I would like to ask , If $G$ be an infinite group , then does there exist a surjection from $G$ ...
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21 views

Irreducible representation of $1$-transposition groups

I would like to know the theory of irreducible representation of $1$-transposition groups. Could anyone provide me a pointer from where I can proceed?
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Definition of $k$-transposition group

In A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture, a $k$-transposition group is defined as follows. Recall that a $k$-transposition group $G$ is one generated by a ...
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23 views

Automorphism groups of partially cycle graphs

I define partially cycle graphs as follows. If we add the same subgraph to $n-k$ vertices of an $n$-vertex cycle graph, where $1\le k < n$, we create a partially cycle graph. Here are a few ...
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25 views

Doubly transitive and solvable

I wonder if there are subgroup of $S_n$ that are solvable and doubly transitive for $n\geq 5$. Can someone find an example or prove that they don't exist?
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1answer
82 views

Least symmetric group having a certain Abelian group as subgroup

Given an Abelian group $G\simeq\bigoplus_{k}\mathbb Z_{p^{n_k}_{k}}$, where $p_1\leq p_2\leq ...$ are primes, how to calculate the least symmetric group $S_n$ having a subgroup isomorphic to $G$?
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2answers
31 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 ...
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1answer
37 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 ...
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34 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 ...
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1answer
42 views

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; ...
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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 $S_{3}$ ...
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1answer
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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
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1answer
72 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, ...
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1answer
41 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 $\frac{n-2}{2} + 4 = \frac{n}{2} + 3$. ...
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1answer
58 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 ...
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1answer
38 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 ...
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13 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 $\...
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1answer
38 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 ...
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1answer
34 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 ...
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0answers
27 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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!
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2answers
31 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 $...
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0answers
23 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 ...
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58 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 ...
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2answers
98 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
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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
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4answers
55 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 &...