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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Maximum order of an element in Alternating group of degree 10.

What is the maximum order of any element in $A_{10}$? My attempt: I tried this problem. But I am not sure about the answer. My answer is 21 because 10 can be written as 7+3. $A_{10}$ can have ...
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1answer
35 views

What does it mean to find elements in $S_9$ that are “not cycles”?

I came across this wording in the following question. Some clarification on what this means and how to approach this problem would be helpful. Thanks!
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2answers
41 views

Help with some simpler symmetric group $S_n$ problems.

I apologize if the problems seem trivial but I have not been able to find example problems or solutions to some of these questions. Could someone please confirm my attempts are correct or not? ...
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1answer
29 views

How to use class equation for determining the center of $S_4$

How to use class equation for determining the center of $S_4$ $$|G|=|Z(G)|+\sum_x [G:C_G(x)]$$ So I guess I need to find $$|G|-\sum_x [G:C_G(x)]=|Z(G)|$$ Well $|S_4|=4!=24$ and $C_G(x)$ is the set ...
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1answer
28 views

transitive subgroups of the symmetric group on $2(d+1)$ elements: can I always do at least $d$ permutations?

I have a set of $2(d+1)$ elements which are labelled as pairs $\{e_i, a_i\}_{i=1}^{d+1}$, transforming under some transitive subgroup of the symmetric group. This can be thought of as a regular ...
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1answer
38 views

On the number of conjugacy classes in $S_n$.

Let $\sigma=(1,2,11)(3,4)(5,6,7,8,9)\in S_{12}$. I am willing to get the number of conjugates of $\sigma$. Clearly if $\tau$ be one such, then it must have the same cycle type. So in other words, we ...
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61 views

The Fifteen Puzzle and $S_n$ [duplicate]

I was studying permutation groups from the book "Abstract Algebra and Applications" by Karlheinz Spindler in which page 553 I came across the following interesting problem. It is on the famous "The ...
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1answer
34 views

Symmetric Polynomials and Automorphisms of Complex Polynomial Rings

I asked a version of this question earlier, but it was very imprecise and poorly formatted, so I decided to create a new question. Suppose we have an ordered set of $n(n-1)/2$ distinct polynomials ...
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1answer
58 views

Product of $n(n-1)/2$ polynomials of the same degree is symmetric

I am trying to prove a simple fact about polynomials in the multivariate polynomial ring $\mathbb{C}[x_1,x_2,...x_n]$, for $n \gt 3$ but I've been getting stuck. EDIT: After a comment by Tad I ...
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1answer
48 views

$SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong \mathbb A_4$

I am trying to prove that the quotient group $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3))$ is isomorphic to $\mathbb A_4$. I could show that $SL_2(\mathbb Z_3)$ has $24$ elements (one can see this ...
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1answer
37 views

Composition of groups

Let's say we have a system of interacting particles that can divided into two populations. The symmetry group of each population is $G$, and the two populations are identical, so that I can exchange ...
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2answers
84 views

How to prove that a given group is isomorphic to Sym(4)?

Given a specific group with 24 elements, I want to prove that it is isomorphic to Sym(4). To begin with, I calculate the orders of my group's elements and they come out as in the order statistics for ...
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1answer
16 views

Need help understanding the precise meaning of “unique factorisation of disjoint cycles”

Below is taken from my linear algebra course lecture notes: Some facts about permutations of $\{1,2,\dots,n\}$: Every permutation is a product of disjoint cycles which commute. For example ...
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17 views

Cycle Structure of a Permutation Based on the Binary Representation

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, ...
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2answers
41 views

Representing a 5-cycle as a product of transpositions

Dr. Pinter's "A Book of Abstract Algebra" shows that: $$(12345)$$ can be written as the following product of transpositions: $$(54)(53)(52)(51)$$ How can the first representation, $(12345)$, be ...
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0answers
23 views

Permutation module $M^\lambda$ as induced module

If we let $r$ be a natural number, $\lambda$ be a partition of $r$, $\Sigma_r$ be the symmetric group on $r$ numbers, we can define the following $K\left[ \Sigma_r \right]$-module: $M^\lambda := ...
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32 views

Conjugation of permutations

In the group $S_n$ I usually use the fact that if $(a_1 a_2 \dots a_r) \in S_n$ is an r-cycle and $\sigma \in S_n$ then $\sigma (a_1 a_2 \dots a_r)\sigma^{-1} = (\sigma(a_1)\sigma(a_2) \dots ...
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32 views

Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
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1answer
22 views

Prove this result relating to the sign of a permutation

Suppose that $\phi \in S_n$ is a permutation. Suppose also that $\psi = \phi \circ (i,j),$ where $1 \leq i, j \leq n.$ Why does it follow that sign$(\phi) = $ $-$sign$(\psi)$?
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Equivalence of irreducible representations of special linear group SL$(n)$ via those of GL$(n)$ and invariant total anti-symmetric tensor

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. Theorem 13.14 discusses the the equivalence of irreducible representations of special linear group ...
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1answer
71 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
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0answers
46 views

Idemptent of Young Tableaux

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. I understood that different Young Diagrams corresponds to inequivalent irreducible representations of ...
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8 views

Modular arithmetic of rotation a triange

A triangle is an n-gon with 3 rotational symmetries and 3 reflection symmetries. Each rotation is 120 degrees. Suppose the triangle begins with initial angle 240 degrees and rotates through 240 ...
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1answer
88 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
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35 views

Trade-off among symmetries

Take a set $X \in \mathbb{R}^2$ of nonzero measure $\mu(X) \neq 0$. I am attempting to design a set that has the following symmetries (continuous or discrete) $1.$ Scale symmetry $2.$ Rotation ...
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2answers
41 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
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32 views

How do I prove that the symmetric group $S_p$ where p is prime can be generated by any transposition and any p-cycle?

I am at a complete loss as to how to even begin. I think it has something to do with the fact that any p-cycle can be represented by $(123...p)$?
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17 views

Why is the k-th convolution of $P_S$ is equal to ${P_S^k}$

For a random walk using transpositions on $S_n$, how can it be explained that the k-th convolution of $P_S$ is equal to ${P_S^k}$. They look to be the same intuitively but how can it written ...
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1answer
65 views

Quotient space of $\Bbb C^n$ obtained by action of $S_n$

Consider the action of $S_n$ on $\mathbb{C^n}$ given by: $$\sigma(x_1, x_2, \cdots,x_n) = (x_{\sigma(1)}, x_{\sigma(2)}, \cdots,x_{\sigma(n)}).$$ What is the quotient space of $\mathbb{C^n}$ obtained ...
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22 views

Check the order of a subgroup of the alternating group $A_n$

Question : For a positive integer $n \geq 4$ and a prime $ p \leq n$. Let $U_{p,n}$ denote the union of all p- syllow subgroups of the alternating Group $A_n$ on n letters . Also let $K_{p,n}$ denote ...
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20 views

why is the sum over even and odd permutations the same?

let $m$ be an $n \times n$ matrix (over $\mathbb{R}$,say) and for a permutation $\sigma \in S_n$ define the monomial: $$ P_\sigma(M) = \prod_{j=1}^n m_{j,\sigma(j)} $$ let $\tau$ be an odd ...
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1answer
25 views

Finding the two different conjugacy classes in $A_n$ after splitting criterion

Suppose we have a group $A_n$ for some $n$ (maybe take $A_5$ as an example). We find the conjugacy classes of $S_n$ which are determined by cycle type. Then we use the splitting criterion ...
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3answers
38 views

Are any two elements, of equal length, conjugate in $A_n$ for $n ≥ 3$

I have given the following problem to solve; (i) Prove any two cycles in Sn of the same length are conjugate in $S_n$ for any $n\geq 3$. (ii) Is the same true in $A_n$ for $n\geq 3$? (iii) Prove ...
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31 views

Generating subgroups of $S_3$

I am trying to understand how the generation of subgroups work: So if I have $\sigma_1= (123)$ and $\sigma_2=(12)$, both of these generate subgroups of $s_3$, how? Do I give it an element, say $1$ ...
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35 views

Non-existence of $\mathfrak{S}_n \hookrightarrow \mathfrak{A}_{n+1}$.

Let $\mathfrak{S}_n$ be the symmetric group (permutations of $n$ items) and let $\mathfrak{A}_n$ be the alternate group. For $n \geq 5$, I have to show that there is no injective morphism ...
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19 views

Does every Young diagram have a unique minimal major index?

Given a Young diagram, $Y_\rho$, corresponding to an irreducible complex representation $\rho$ of the symmetric group $S_n$, we can associate a set of major indices $\{ ...
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1answer
21 views

If $\sigma\in S_p$ with $|\sigma| = p$, why is $\sigma$ a $p-$cycle, and why is $|\sigma^r| = p$?

If $\sigma\in S_p$ with $|\sigma| = p$, why is $\sigma$ a $p-$cycle, and why is $|\sigma^r| = p$ for each $r$, $1\leq r < p$? ($p$ is a prime) I guess I am just having a hard time understanding ...
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1answer
32 views

Commutators of a symmetric group

I am trying to prove that the commutator subgroup of $S_n$ ($S_n$ is a symmetric group on $[n]$), $[S_n,S_n]$ consists solely of commutators $s_1^{-1}s_2^{-1}s_1s_2$ for some $s_1,s_2\in S_n$. Any ...
2
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0answers
22 views

Composition of Polynomials and Galois Theory

Let $f(x)$ be a polynomial of degree $n$ over $\mathbb{Q}$, with Galois group isomorphic to the symmetric group $S_n$. How do I show that $f$ cannot be expressed as a composition $g(h(x))$ of two ...
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1answer
81 views

Let $G$ be a finite group, $p$ the smallest prime divisor of $|G|$, and $x\in G$ an element of order $p$.

Suppose $h\in G$ is such that $hxh^{−1}=x^{10}$. Show that $p=3$. I am trying to solve this problem using group actions. Let $H$ and $X$ be the subgroups of $G$ generated by the elements $h$ and $x$, ...
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1answer
51 views

How many solutions of equation

How many solutions of equation $x_1+x_2+x_3+x_4=n$ in $N_0$ such that $x_1\leq x_2\leq x_3 \leq x_4$? I found solutions of $x_1+x_2+x_3=n$ in $N_0$ , $x_1\leq x_2\leq x_3 $ in the following way : ...
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1answer
25 views

Proof the isomorphism between symmetric group & subgroup ?

For the symmetric group S(2k) there are two equal subsets A = {1,....,k} and A' = {k+1,.....,2k}. Let L be the subgroup of all permutations r of S(2k) with r(A) = A or r(A)= A' and r(A') = A or r(A') ...
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27 views

Geometric understanding of why $D_1\cong \mathbb{Z}_2$

From what I understand, $P_2$ is the line joining $(-1,0)$ to $(1,0)$ and then $P_1$ is the point $(1,0)$. This is due to defining $P_n$ (the regular polygon with # of points $n$) to be formed by the ...
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2answers
69 views

Jucys-murphy elements commute with each other.

In the group algebra $\mathbb{C}[S_n]$, for $1<i<j\le n$, $X_i=(1\ i)+(2\ i)+...+(i-1\ i)$ and $X_j=(1\ j)+(2\ j)+...+(j-1\ j)$ commute with each other. I have been trying to do it ...
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1answer
37 views

In the Group $(S_{3},\circ)$, what are the elements of the group $\Big(\big((123)\big), \circ\Big)$?

Given the Group $(S_{3},\circ)$ What are the elements of the group $\Big(\big((123)\big), \circ\Big)$? Also, why does $\big((123)\big)$ have two brackets around it?
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1answer
38 views

Transposition as generating $S_n$.

As far as I know the group of symmetries of the euclidean n-cube is generated by reversions ${r_i}$ and transpositions ${s_i}$ for i=1,...,n-1. Transpositions generate the subgroup of permutations of ...
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1answer
27 views

basis for symmetric square

If we have the symmetric square Sym$^{2}V$ and $V=\mathbb{C}^{2}$, why is it that $\{x^{2}, xy, y^{2}\}$ form a basis for it? So symmetric square matrices are when the main diagonal acts as a ...
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20 views

Multiple maximal cyclic subgroups of a symmetry group

If a symmetry group T has a maximal cyclic subgroup Cn because of a projection I1, then it means it will have a rotational symmetry order of n. If we have another projection (of the same object) with ...
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1answer
38 views

How can we prove that SO(2) is a subgroup of SO(3)

I have a fixed plane that takes the projection of a 3D image and we need to prove that all the rotations, fixing the plane, is a subgroup of SO(3). From basic understanding I know that the ...
3
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2answers
56 views

Normalizer of the cyclic group in $S_n$

Let $G = S_n$ and $H = \langle (1,2,\ldots,n) \rangle.$ It is not too hard to see that $$C_G(H) = H.$$ What I am now wondering is, which group is $N_G(H)?$ Is there any way to determine that? I ...