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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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
29 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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18 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
55 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
21 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 ...
2
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2answers
66 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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36 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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35 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
21 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
31 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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2answers
45 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
49 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
19 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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20 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
44 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
42 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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43 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
41 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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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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454 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
18 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
215 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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32 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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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
38 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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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} ...
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2answers
45 views

Composition of two rotations of the same angle $\alpha$ fixing points $a,b \in{S^2}$

Let $g,h$ be rotations of the same angle $\alpha$ around fixed points $a,b \in{S^2}$. Show $gh$ fixes $c \in{S^2}$ on the great circle that forms the perpendicular bisector of the segment $ab$, such ...
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35 views

If a normal subgroup of a symmetric group contains a transposition. Prove that the normal subgroup is the entire symmetric group.

Here is how I have done this problem. I would appreciate if someone can comment on my proof. Let $N\triangleleft S_n$ and transposition $(a_i b_i) \in N$ where $a_i,b_i\in (1,2,....,n)$ Then ...
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29 views

If $a$, $b$ are rotations of $S^2$ through the same angle, show that they are conjugate in $SO(3)$.

Basically everything to do with the question is in the title! I'm just not really sure where to begin with this question, as I don't want to get bogged down in complicated matrix multiplication. I'm ...
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31 views

Different forms of lagrange theorem

I know the proof of the original Lagrange Theorem which states that: Given a group $G$ the order of its subgroup $H$, divide its order. There are two more statements related that I do not know ...
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37 views

symmetric group acting on torus

Let $S_k$ be symmetric group of order $k$. Let $T^k=S^1\times\cdots \times S^1$. Then $T^k$ is a Lie group. For each $\sigma\in S_k$, let $\sigma$ act on $T^k$ from right in the way $$ ...
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40 views

Prove that group of symmetries is isomorphic to $S_n$

In my algebra book the first section has the following exercise: Prove that group of all symmetries(isometric bijections under composition) of a regular tetrahedral is isomorphic to $S_4$. I did it ...
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2answers
39 views

Finding the inverse of an element of $S_n$ and it's order [duplicate]

I have two questions, 1) What are the ways to find the inverse of an element of $S_n$? 2) What are the ways to find the order of an element of $S_n$?
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1answer
83 views

Finite groups with nontrivial outer automorphisms

Let $G$ be a finite group whose center $Z(G)$ is trivial. Suppose that the group $\text{Out}(G)$ of outer automorphisms is nontrivial. Question: Does there always exist an $f \in \text{Aut}(G)$ ...
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Reference on the Crystallographic restriction theorem and some related results

Firstly I precise that I am working on $\mathbb C$ the plan of complex numbers I have some result for which I look for references (mathematical books or articles) where the reader can find their ...
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25 views

What is the best way to look at the symmetric group $S_3$in relation to a triangle?

For the group $S_3$, we have 6 elements or functions(the upper row is the input, the lower row is the output: $\rho_0=\begin{pmatrix}1&2&3 \\ 1&2&3\end{pmatrix},\ ...
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28 views

Finding $\gamma \in S_7$ that satisfies $\gamma^4=(3412675)$

I have proved it using an arbitrary $\tau=(1234567)$, taking it to the fourth power, and finding that $\tau^4=(1526374)$. Can I just see the pattern of where each element went and match it to the ...
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35 views

Find the centraliser of $(12)(34)$ in $S_4$.

I am trying to find the centraliser of $(12)(34)$ in $S_4$. I have that $$\begin{align}C_{S_4}((12)(34)) &=\{g \in G : g(12)(34)g^{-1}=(12)(34) \} \\ &= \{g \in G : (g(1) ...
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59 views

Geometric Interpretation of S3

My impression was that the symmetric group $S_3$ acts on the vertices of a labeled triangle. However, I am not sure this is the case anymore, because of the following. (The triangle is labeled as ...
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2answers
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Exercise 1: Galois Theory (J. Rotman)

Definition: Let $F$ a figure in the plane, its symmetry group is defined by $\Sigma(F):=\{\sigma \in O(2,\Bbb R)\mid \sigma(F)=F\}$. Here $O(2,\Bbb R)$ denotes the real orthogonal group. Exercise 1: ...
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Eigenvalues of operator on $S_n$'s group algebra

Take the group algebra of the symmetric group $S_n$ (or equivalently consider $S_n$'s regular representation) - I guess over $\mathbb{C}$. If $e_{i,j} \in S_n$ denotes the element which swaps only ...
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1answer
35 views

Boundedness of a modular form in $\mathbb{H}$

Let be $k>0$ and $f \in S_k(\Gamma)$. I want to show that the function $h(z)=Im(z)^{\frac{k}{2}}\cdot |f(z)|, \; z\in\mathbb{H}$ is bounded in $\mathbb{H}$. I have already shown, that $h$ is ...
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29 views

Cuscs of a subgroup of $\Gamma$

I'm going to be completely honest about this: I need the solution of this to get permitted to the exam in complex analysis. The topic is not even relevant for the exam and I am absolutely not able to ...
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35 views

Exercise on representations of the Dihedral group (Etingof 3.17)

I'm confronted once more with a problem on representation theory which I cannot fully solve (Problem 3.17 http://math.mit.edu/~etingof/replect.pdf): Let $G$ be the group of symmetries of a regular ...
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1answer
57 views

What are the irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$? [closed]

Find with proof all irreducible representations $V$ for $S_n$ over ${\bf C}$ that admit a nonzero vector fixed by $S_{n-1}$.
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45 views

How many squares in a finite group?

Let G be a finite group and denote by S[G] the number of squares in G. The maximum, S[G]=n, is attained for a group of odd order n since each element has a square root in that case. At the other ...