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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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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1answer
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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
23 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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0answers
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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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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
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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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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 ...
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1answer
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cohomology of permutation group with mod 2 coefficient

Let $S_n$ be the permutation group of order $n$. Let $\mathbb{Z}_2=\mathbb{Z}/2\mathbb{Z}$. What is the cohomology algebra $$H^*(S_n;\mathbb{Z}_2)?$$ For $n=2$, $BS_2=\mathbb{R}P^\infty$ hence I ...
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1answer
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Do involutions in the symmetric group form a basis of a Jordan algebra?

Let $M_d$ is a vector space spanned by the set of all the involutions in the symmetric group $S_d$ (you can treat $M_d$ as a space of function on the set of involutions endowed with standard addition ...
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1answer
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Find permutation $B$ given $B^4 =(2143567)$

Let $ B\in S_7 $ and $ B^4 =(2143567)$. Find B. How to find $B$? All I know is that $B^7 $ is identity permutation because it is a 7 cycle.so (B^4)^2 should be B?
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1answer
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calculate the number of sylow p subgroups of a5

Calculate the number of Sylow $p$-subgroups of $A_5$ We have $|G|=60=2^2\cdot 3\cdot 5$ Let $n_p$ be the number of Sylow $p$-subgroups of $G$. By Sylow's third theorem, we have $n_3\in\{1,4,10\}$. ...
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$n-1$ dimensional permutation module for $S_n$

Say $n \ge 5$. Let $P$ be the $(n-1)$ dimensional permutation module for $S_n$, i.e. the permutation representation on $\{(x_1, \dots, x_n) \in {\bf C}^n: \sum x_i = 0\}$. Prove that: $\wedge^2P$ ...
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0answers
32 views

Symmetric group $S_4$ [duplicate]

Let $G$ be the Symmetric group $S_4.$ Give a representative of each conjugacy class of $G.$ Then calculate the size of each conjugacy class. I have no idea how to do this.
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Relationship of irreducible polynomial of prime degree $p$ and the full symmetric group $S_p$

According to Wikipedia, if $f(x)$ is an irreducible polynomial of prime degree $p$ over $\mathbb{Q}$ and it has two nonreal roots, then the Galois group of $f$ is the full symmetric group $S_p$. For ...
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1answer
31 views

Symmetric Group acting on $X \times X$

The symmetric group $S_n$ acts on the set $X = \{1,\ldots,n\}$ and hence acts on $X \times X$ by $g(x,y) = (gx, gy)$. Determine the orbits of $S_n$ on $X \times X$. Not sure how do I actually ...
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2answers
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Splitting of conjugacy class in $A_n$

During reading, I have encountered this, in several places: The following are equivalent for a permutation $\sigma \in A_n$: 1) the $S_n$-conjugacy class of $\sigma$ splits into two $A_n$-classes 2) ...
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1answer
30 views

Permutation inverse form

Given: $A=\{1,2,3,4,5,6\}$, $P_1=\begin{pmatrix} 1 &2& 3& 4& 5& 6\\ 2& 3& 4& 1& 5& 6\end{pmatrix}$, $P_2=\begin{pmatrix}1 &2 &3 &4& 5 ...
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1answer
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Subgroups of $S_n$ that can send any subset of $[n]$ to any equally sized subset of $[n]$

This is a repost of a question I was trying to solve yesterday that got deleted. The question asked for a characterization of the subgroups $G$ of $S_n$ which when endowed with their natural action on ...
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Determine all the normal subgroups of $S_4$ [duplicate]

Previous part: Give a representative of each conjugacy class of $G$ - $e,(1,2),(1,2,3),(1,2,3,4),(1,2)(3,4)$ Calculate the size of each conjugacy class of $G$. For: $e-1$, $(1,2)-6$, ...
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1answer
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stabilizers and orbits of elements in $S_4$

Let $S_4$ be the group of symmetries of $4$ letters, and let $S_4$ act on itself by conjugation. Let $\sigma = (12)$, $\tau = (123)$, and $\rho = (1234)$. Find the stabilizers of the elements ...
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Writing down elements using cycle notation.

I have $GL_2({\Bbb{Z}}_2)$ in which ${\Bbb{Z}}_2$ consists of the integers $\{0,1\}$. We observe that $|M_2({\Bbb{Z}}_2)|=2^4$. Now let's define Y to be the set of all non-zero elements of ...
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1answer
25 views

Checking if an element of a certain order is present in $S_n$

This exercise asks to provide elements of order 10, 20, and 30 in $S_{10}$. Thinking that the order of a permutation $\sigma$ is the least common multiple of the length of the disjoint cycles whose ...
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1answer
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Is the question phrased properly? and is my proof correct? (An infinite alternating group is simple)

I'm interested in the following exercise from Dummut & Foote's Abstract algebra text (p. 151) Let $D$ be the subgroup of $S_\Omega$ consisting of permutations which move only a finite number ...
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1answer
45 views

Is my proof correct? ($A_n$ is generated by the set of all 3-cycles for $n \geq 3$)

I want to prove that for $n \geq 3$, the alternating group $A_n$ is generated by the set of all 3-cycles. Here is my attempt: Let $\mathcal{S}$ be the set of all 3-cycles in $S_n$, which is a ...
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Intuition behind the construction of Young Symmetrizer

I've been studying representation theory of group on Tung's "Group Theory in Physics". I understood Young Symmetrizers of different Young diagrams are essentially primitive idempotents in group ...
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Sylow subgroup of $S_{11}$

I want to construct some Sylow $3$-subgroup of $S_{11}$.This subgroup has $3^4$ elements. I know any Sylow $3$-subgroup is isomorphic to $(\mathbb{Z}/3\mathbb{Z})^3\rtimes P$ where $P$ is a Sylow ...
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1answer
36 views

proving a group

Consider the group ($[0; 2\pi)$;$\oplus_{2\pi}$) where $\oplus_{2\pi}$ means addition modulo $2\pi$. Define $S_1$ as the following subset of $\mathbb{C}$ $S_1$ = {$z \in \mathbb{C}; z = e^{i\theta}; ...
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1answer
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Proof of the fact that the set of (p,q) shuffles is a cross section of the subgroup $S_p\times S_q$

Definition Let $G$ be a group and $H$ its subgroup. We name a subset $K$ of $G$ a cross section if it has exactly one element from each left coset of $G/H$. Definition Let $n=p+q$ for some ...
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If $H<S_n$, $H$ is abelian and transitive on $\{1,2,…,n\}$, then the order of $H$ is $n$ [duplicate]

If $H<S_n$, if $H$ is Abelian and transitive on $\{1,2,...,n\}$, then the order of $H$ is $n$. So far I have: $H$ is transitive therefore the group orbit is the group itself. I know I should ...
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The structure of the Sylow p-subgroups of $Sym(n)$

For the structure of the Sylow $p$-subgroups of $Sym(n)$, there is a standard proof by using the properties of the wreath product like as in the Passman's book. But I want to understand this proof. In ...
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1answer
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Non-simplifiable permutation matrices

The permutation matrices for 2 and 3 dimensions look like this: 2-dimensional: $$\quad M_1^{2d}=\left(\begin{matrix}1 &0\\0 &1\end{matrix}\right), M_2^{2d}=\left(\begin{matrix}0 &1\\1 ...
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1answer
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Show $A_6$ is simple

I have to show that the group $A_6$ is simple. For the subgroups that have order divisible by $5$ and order of $8,9,18,24,36,$ and $72$ I have shown that those subgroups are not normal. Now we need to ...
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Showing product of disjoint cycle

I am trying to show the product of two disjoint cycles such that they have nothing in common for $A_n$ for $n\ge 3$. So I have the two cycles $(ab)(cd)$. I have read here: ...
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Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
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two books on symmetric groups

Listed below are two books on new approaches to the representation of symmetric groups. Could anyone explain to me the difference/connection between these two approaches? which approach will be more ...
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1answer
35 views

Combinatorial puzzle concerning labelled equilateral triangles

Consider equilateral triangles $\Delta$ of fixed size and in a fixed position with each side labelled by a label $l \in \{1,\dots,k\}$. Obviously there are $k^3$ such labelled triangles. Let ...
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1answer
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Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
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1answer
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When p is prime number, explain all the elements of $S_p$, whose the order is not divisible by any prime numbers less than p

I'm studying a first course in abstract algebra, and currently I'm stuck with this problem. The problem is that, when p is prime number, explain all the elements of $S_p$, whose the order is not ...
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3answers
135 views

Proof that $A_n$ the only subgroup of $ S_n$ index $2$.

I have what seems to me a very simple proof that $A_{n}$ is the only subgroup of $S_{n}$ of index 2. Since I've seen other people prove it with what feel like really complicated methods (Like here.), ...
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Conjugation of $A_5$ by $S_5$ and effects on irreducible representations of $A_5$

I want to ask q5 of the following https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2010-2011/repex2.pdf The group $S_5$ acts on $A_5$ by conjugation. How does this action act on the ...
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permutation representation of Symmetric group

The symmetric group $S_n$ acts on $\mathbb{C}^n$ by permuting the coordinates. Decompose this representation explicitly into irreducible representations. For the action $\sigma (a_1,\cdots, a_n)= ...
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Prove that $S_n$ is a group with respect to composition

Prove that $S_n$ is a group with respect to composition $fn = f\text{ follow }n$. I know that to prove something is a group I have to show identity, inverse, close, and associativity. But I do ...
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1answer
37 views

Factor Group, Isomorphism

If group $A=S_3⊕\mathbb{Z}_4$ and subgroup $B=\langle (132),2\rangle$, find a group the factor group $A/B$ is isomorphic to and construct the group table for $A/B$. I'm really not sure what to do with ...
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In a group of symmetries, find elements such that $\sigma \circ\beta = \beta \circ\sigma$ and $\sigma \circ\beta = \beta^{-1} \circ\sigma$

(i) Let $\sigma = (12)(345) \in S_5$. Find all $\beta \in S_5$ such that $\sigma \circ\beta = \beta \circ\sigma$. $S_5$ is the symmetric group of 5 elements. (ii) Let $\sigma \in S_n$. Show that ...
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1answer
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Irreducible representations of $S_3$

I am trying to do a problem from artin's algebra 2nd ed (Chapter 10, Exercise 2.3) but having trouble: Let $(\rho , V)$ be a representation of the symmetric group $S_3$. Let $x=(123), y=(12)$ be the ...
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1answer
25 views

subgroups with cycles

Let Sym(n) denote the symmetric group on n letters and let H be a subgroup of Sym(n) . Suppose that H contains a k cycle for each value of k from 2 through n . This should be enough to ...
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1answer
59 views

Representation of the symmetry group (rotations) of the icosahedron

Suppose $I$ is the set of vertices of the regular icosahedron, here is a link of the icosahedron: http://www.werheit.mynetcologne.de/icosaeder.gif Let $F(I)$ be the space of complex functions on $I$, ...
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1answer
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For $H$ fixed point free show $|H| \leq n$

Suppose that $H \leq S_n$, and suppose that $H$ has the property that all non-identity elements of $H$ are fixed-point free. Show that $|H| \leq n$. I am trying to prove this by induction. For ...
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2answers
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Problem on symmetric group of order $4$

Let $S_4$ denote the group of permutations of $\{1,2,3,4\}$ and let $H$ be a subgroup of $S_4$ of order $6$ Show that there exists an element $i$ in $\{1,2,3,4\}$ which is fixed by each element of ...
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77 views

What is the order of the alternating group $A_4$?

When I write out all the elements of $S_4$, I count only 11 transpositions. But in my text, the order of $A_4$ is $12$. What am I missing? ...