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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21 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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38 views

Find all Lie point symmetries generators for nonlinear equation [closed]

Find all Lie symmetries generators for nonlinear equation $$u_t=\frac {-1}{u_{xx}}$$
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67 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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33 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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7 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
81 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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32 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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39 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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22 views

Describing the order in symmetric group.

Once again I got stuck in my work. I'm not sure how to understand these conditions, let alone trying to figure out the proof. Any explanation will be greatly appreciated. Next we define a partial ...
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25 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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16 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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20 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
19 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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33 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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31 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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17 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
19 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
30 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 ...
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20 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
69 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
48 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
24 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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26 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
68 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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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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37 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
24 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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18 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
34 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 ...
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51 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 ...
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1answer
50 views

Can two different characters of $S_n$ have the same _multiset_ of values?

As I was going through various representation-theory posts in the site, I stumbled upon this one: Characters of the symmetric group corresponding to partitions into two parts. Now, that question ...
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1answer
22 views

Sylow $p$-subgroup of a normal group.

Let $G$ be a transitive subgroup of $S_p$ and let $H$ be a non-trivial normal subgroup of $G$. I need to show that any Sylow p-subgroup of $G$ is also contained in $H$. I know that any transitive ...
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23 views

Sylow $p$-group of a transitive subgroup of $S_p$

How do I show that any transitive subgroup of $S_p$ contains a non-trivial Sylow $p$ subgroup, of cardinality $p$? I am trying to prove a result of Galois and the only hint I have is that if $p$ is a ...
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3answers
45 views

$S_4 \ne \langle (1,2,3,4), \, (1,3)\rangle$

So I'm trying to prove $S_4≠⟨(1,2,3,4),(1,3)⟩$, and I get the basic idea that $(1,2)$ swaps two things next to each other, which neither of the other operations do, and necessarily neither do their ...
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1answer
45 views

About conjugacy in $A_n$

I know that $\sigma$, $\tau$ $\in$ $S_n$ are conjugate if and only if they have the same cycle structure. Is there any explicit way that we can determine whether two elements in $A_n$ are conjugates? ...
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48 views

Why is the permutation $(a,c,d,e)(a,b)=(a,d)(b,c,e)$

I'm working through a proof in my notes. We already know that the transposition $(a,b)\in G$ and $(a,b,c,d,e)\in G$, where $G$ is a group of permutations of the elements $a,b,c,d,e$, so it's a ...
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1answer
43 views

Are there two isomorphic subgroups of the symmetric group of a countable set which are not conjugate

In the following questions we are dealing with subgroups of SYM($\aleph_0$) (the group of permutations of a countable set) with each non identity element having infinite support. 1. Are there two ...
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1answer
24 views

Abstract Algebra Symmetric Groups

$$ \begin{align} \beta &= \begin{bmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ 1 & 3 & 8 & 7 & 6 & 5 & 2 & 4 \end{bmatrix} \\ &= ...
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468 views

The Weaver Android app $\rightarrow$ cute combinatorics problem

There's an Android puzzle app called "The Weaver". My question is why every level seems to be solvable in far fewer moves than one might naively think. Here's a link for people who want to play along ...
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1answer
19 views

Does there exists a proof that any divergentless tensor can be decomposed into the sum of divergentless symmetric and antisymmetric tensors?

A friend and I attempted to work out the proof on the board that any divergentless asymmetric tensor can be written as the sum of divergentless symmetric and antisymmetric tensors. We wrote down the ...
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1answer
219 views

The sum of orbit size of some element over the image of group “polynomial”

$\DeclareMathOperator{\orb}{orb}$ Say I have a group "polynomial", $p$, on $S_n$, that is $p(x)=a_1 x^{\epsilon_1}...a_n x^{\epsilon_n}$ for all $x \in S_n$, fixed $a_i \in S_n$ and fixed $\epsilon_i ...
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35 views

Question about permutations: How to show $\sigma(P)=(-1)^{\imath(\sigma)}P$?

A permutation of a finite set $X$ is any bijection from $X$ to $X$. We denote by $S(X)$ the set of all permutations of $X$. If $I_n:=\{1, \ldots, n\}$ we write $S_n$ instead of $S(I_n)$. Define ...
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1answer
66 views

Every group as full symmetry group of points in $\mathbb R^d$

Does every finite group $G$ have the property that it is isomorphic to a full symmetry group of some set of points in $\mathbb R^n$ for some $n$
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37 views

Name of transitive group of polynomial with cubic degrees

What is the transitive group details of a polynomial where only the third power terms occur? That is $x^{3n} + a_{n-1} x^{3(n-1)} + ... + a_1 x^3 + a_0$. I need the basic theorems that state or ...
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1answer
43 views

Symmetries of the regular hexagon

Q- Let G be the group of the symmetries of the regular hexagon. List the elements of G (there are 12 of them), then write the table of G. So for the listing the elements of G, they want it like this: ...
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46 views

Integrating over a symmetric-group function (elements being permutations)

I would like to integrate a permutation of a function. Namely I have the following: $\sum_{\sigma, \sigma'\in S_{n+1}}\int_{-A}^A dz_1dz_2 ... dz_{n+1} ...