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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Find commuting elements within a permutation group

The question is like this: IF $G=S_5$ and $g=(1\quad 2\quad 3)$, determine the number of elements in $H=\{x\in G:xg=gx\}$. To do the question, first it says $$x(4)=(x(1\quad 2\quad 3))(4)=(1\quad ...
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Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
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18 views

Cycle structure of the generators of the dihedral group

Would the following be correct about generating the dihedral group $D_n$ by permutations? If $n$ is even, the group can be generated as $\langle(2\quad n)(3 \quad n-1) \ldots (\frac{n}{2}-1 \quad ...
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characters in semi-direct product.

The character tables of the irreducible representations of $T_d$ and $C_{3v}$ are linked. In the notation on those pages, $A_1$ and $A_2$ are irreducible representations of degree 1, $E$ is degree 2 ...
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16 views

Why doesn't the coset (1,4,2,3)K belong to the Quotient group

I had been given the following question and answer ( in the image) However i do not understand, why for example: (1,4,2,3)K does not belong the the quotient group? Is there any faster way of ...
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Counting the number of “distinct” permutations of two sets?

I don't really know how to introduce this question, so I start defining something I needed in order to well understand the problem I met! Let $A$, $B$ two finite sets of distinct elements, with ...
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1answer
29 views

Group action and equivalence relation

Let $G$ be finite, and group action on $X\subseteq G$: $g\cdot x:=g^{-1}xg$. Let $G=S_n$, and $X=S_n.$ Show that $[x]_R$ consists of all elements of $S_n$ that are of the same cycle-type as $x$. I ...
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Vertex-transitivity of the automorphism group of a digraph

I am trying to understand the theorem 3 of Cycles in graphs and groups by Kantor. Theorem $3$ If $G$ is a vertex-transitive group of automorphisms of a digraph $\Gamma$ with outdegree $d \ge 1$, ...
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53 views

Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...
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1answer
12 views

Count all the function f satisfying followings : [duplicate]

For $A=\{1,2,3,4,5,6,7,8,9,10\}$ Define a function $f : A\to A.$ Then 30 times composite of f, that is ; $f\circ f\circ...\circ f(x) = x$ and 30 is the least number for f to become an identity. How ...
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49 views

How many number of functions are there?

$A=\{1,2,\dots,10\}$ Define $f:A \rightarrow A$ then $f^{30}(x) = x$ ($30$ times composite of $f(x) = x$ and the number $30$ is the least number for $f$ to become an identity function) How many ...
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25 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
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41 views

What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
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29 views

Normality is not transitive

Let $G=S_3\times S_3$ where $S_3$ is the symmetric group. Let $p= \begin{pmatrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ \end{pmatrix} $, let $L=(p)$, $K=L\times ...
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Finding the character table for Z_8

I am a bit confused about how to come up with the number of irreducible representations, as well to come up with the number of different conjugate classes. Starting me out would be highly appreciated ...
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1answer
37 views

Number of Elements in a Conjugacy Class of $S_N$ (Derivation)

Consider the conjugacy classes of the symmetric group $S_N$. Each conjugacy class consists of permutations that have the same cycle structure. We see that the number of possible cycle structures is ...
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Fixed-point subspace of $O(2)^-$, a subgroup of $O(3)$

$O(2)^-$ is generated by the $SO(2)$ of rotations about the $z$-axis and a reflection through a vertical plane. The space $V_l$ is generated by spherial harmonics, i.e., Cartan decomposition ...
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Decomposition of permutation

I was asked to decompose the permutation $$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 3 & 4 & 5 & 1 \\ \end{pmatrix} = (12345) \in S_5$$ into a product of two ...
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Commutative diagram for hidden subgroup representation of graph automorphism

The hidden subgroup representation of the graph automorphism problem is defined in the section 10.2 of QUANTUM ALGORITHMS FOR PROBLEMS IN NUMBER THEORY, ALGEBRAIC GEOMETRY, AND GROUP THEORY. It is as ...
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29 views

Permutations of $S_7$

Find all permutations $\alpha \in S_7$ such that $\alpha^3 = (1 2 3 4)$. My attempt: We know that such an $\alpha$ must "look like" $(1432)$, since $(1432)^3=(1234)$. I think I need to find the ...
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30 views

Nonconjugate elements of $A_n$ with the same cycle type

I know that in $S_n$, two cycles are conjugate if and only if they have the same cycle structure. This isn't true of $A_n$ though because apparently $(123)$ and $(213)$ aren't conjugate in $A_3$. My ...
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1answer
48 views

Basic Symmetric Group Question

Let $$σ = \begin{pmatrix}1 &2 &3 &4 &5 &6 &7 &8 &9 &10\\ 3 &4 &2 &8 &5 &7 &6 &10 &1& 9 \end{pmatrix} $$ Find $σ^{2345}$ and ...
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Decompose induced representation of $S_2$ and $S_3$

Let $ H=S_2 \subset G=S_3 $. Then use Frobenius reciprocity to decompose $ \operatorname{Ind}_H^G(\operatorname{sgn}_H) $ into irreducibles. $ G=S_3 $ has $ 3 $ irreps $ 1_G, ...
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Mackey's criterion and double cosets of $A_3$ and $S_3$

State Mackey's criterion $Ind_{H}^{G}$ is irreducible $\iff p$ is irreducible $p^s$ and $p$ are disjoint representations of $H \cap sHs^{-1}$ for any $s \in T $\ $ \{1\}$ Find the double ...
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Show $Res_H^G(sgn_G)=sgn_H$ where $G=S_4$ and $H=S_3$

Let $ G=S_3 $ and $ H=S_2 $. Show that $ Res^G_H(sgn_G)=sgn_H $ The symmetric group $G=S_3$ has three irreducible representations $ 1_G, sgn_G $ and $ V$ where $ 1_G $ denotes the trivial ...
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1answer
22 views

Does $A_n$ split a complement of the stabilizer of a partition of $[n]$?

Let $G=S_n$ be the symmetric group on $[n]$, and consider its natural action on the set of partitions of $[n]$. (I mean set partitions, not like in number theory.) Let $\pi$ be a partition and let ...
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25 views

Inverse Galois Theory - Proofs for the symmetric and alternating group.

I am currently looking for paper showing that the symmetric and alternating groups can be represented as Galois groups of polynomials (with rational coefficients) but I'm having trouble finding it. ...
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1answer
80 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
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1answer
39 views

Permutation matrix homomorphism

Can someone please help me prove that permutation matrix is homomorphism? By that, I mean, let $f: S_n \to GL_n (\Bbb R), f(\sigma)=A_\sigma$ is homomorphism. The book tells me to prove it myself I ...
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Transitive action of a $p$-group on minimal block systems

I have trouble proving the following theorem: Let $P$ be a transitive $p$-subgroup of ${\rm Sym}(A)$ with $|A| > 1$. Then any minimal $P$-block system consists of exactly $p$ blocks. Furthermore, ...
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Product of Cycles: Name to denote “direction” of composition

Is there a notation to denote the difference between these two products of cycles? It seems as though there are two conventions out there that should have a specific name for them. The subscripts for ...
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1answer
33 views

Questionable proof, centre symmetric group

I tried proving that for all $n \in \mathbb{N}_{>2}$ the centre $Z(S_{n})$ of the symmetric group $S_{n}$ is trivial. I already have a proof, but I'm not actually sure whether it's correct. Would ...
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61 views

Can $S_n$ be a cyclic group?

Some notes before the question: 1- there are many questions in MSE asking about elements generating $S_n$ but they all involve more than one transposition to generate $S_n$ for example "$S_n$ is ...
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Calculating amount of $2$-sylow subgroups of $S_{2^n}$.

Main question: How do I calculate the number of $2$-sylow subgroups of $S_{2^n}$? Let $n \in \mathbb{Z}_{\geq 2}$. I have a $2$-sylow subgroup $H \subset S_{2^n}$ (too long to spell out all the ...
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$\mathbb{R}^{N}/\Sigma_{n}$ as a topological space

Let $\Sigma_{n}$ denote the symmetric group on $n$ letters. $\Sigma_{n}$ acts on unordered pairs $\{i,j\}$ via $\sigma(i,j)=\{\sigma(i),\sigma(j)\}$. Let $e_{\{i,j\}}$ be a basis for $\mathbb{R}^{N}$ ...
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presentation of the symmetric group via transpositions fixing one element

Consider the symmetric group $S_n$. If we use the most popular set of generators $\sigma_1, \sigma_2,\cdots,\sigma_{n-1}$ with $sigma_i$ being the transposition $(i \, i+1)$, it is well known that ...
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Find the elements of $S_6$ that commute with $(1234)$ [duplicate]

Let $σ=(1234)∈S_6$. List all elements in $N(σ)={α∈S_6∣σα=ασ}$. I know that this is related to cycles and orders, but I am having trouble finding the order of $N(σ)$ initially. I also know that $σ$ ...
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25 views

Why is vertical flip not in the symmetric group of isosceles triangle?

Why is flip around the horizontal axis (2 times) not in the symmetric group of isosceles triangle? Many sources say that there is only a $0^°$ rotation and a flip around the vertical axis.
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Procedure to construct a map from the automorphism group of a graph to the natural permutation representation

Let $\Gamma$ be a graph with $n$ vertices. Let $\varphi_\Gamma$ be the map from the symmetric group $S_n$ to the space of natural permutation representation $\text{Mat} \left(n, \mathbb{C}\right)$ ...
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Do elementary permutations generate an infinite symmetric group?

I am illiterate in algebra/set theory, so I hope you will forgive my ignorance. It is known that the symmetric group of a finite set is generated by elementary permutations. More precisely, ...
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1answer
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problem with alternating group with order 3

I was doing some computation with $A_3$, the alternating group on 3 letters. I know it has to be abelian, even cyclic, but when I carried out the actual computation... I couldn't make sense out of it. ...
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nth power symmetric polynomial in terms of Schurs polynomial

The Schur's polynomial forms the basis of the symmetric algebra so does the power symmetric function. nth power symmetric function are the function of the form $\sum_i x_i^n$. Let $\lambda \vdash n$ ...
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The set of all even permutations in G forms a subgroup of G

Show that if $G$ is any group of permutations, then the set of all even permutations in $G$ forms a subgroup of $G$. I know that I need to show the closure, identity, and inverses properties ...
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1answer
51 views

If $G \to G'$ are isomorphic groups, then $|Aut(G)| = |$ all isomophisms $G \to G'|$.

I'm having trouble proving this. If $G$ is finite (which is not given) with $|G| = N$, then $Aut(G)$ can be seen as a symmetry group with order $N!$, so there are $N!$ possible isomorphisms $G \to ...
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Prove that every permutation in $S_k$ is the product of transpositions of the form $(j, j + 1).$

Prove that every permutation in $S_k$ is the product of transpositions of the form $(j, j + 1).$ I proved the case $n=2$ for my base case... so $(12)=(21)$ and $(21)=(12)(12)$ then I proved $n=3$ and ...
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Is it true that when $n \geq 5$ there is a surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$? [duplicate]

Is it true that when $n \geq 5$ there is a surjective homomorphism from the symmetric group $S_n$ to $S_{n-1}$? How come this is so? Does it have to deal with the subgroup $A_n$ being simple?
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Sylow $2$-subgroup of $A^m \rtimes S_m$

Let $G=A^m \rtimes S_m$ where $A$ is some abelian group. Now what can I say about sylow $2$-subgroup of $G$. The text I am reading says let $S$ be the fixed sylow $2$-subgroup $S_2 \rtimes S(12)$. ...
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Compute $\Sigma_{\pi \in S_n} f(\pi)$ and $\Sigma_{\pi \in S_n} f(\pi)^2$.

Suppose $V_{def}, V_{std}, V_{triv}$ are the defining , standard and trivial representations of the symmetric group $S_n$. And let $V_{def} \cong V_{std} \oplus V_{triv}$, and suppose the characters ...
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Disjoint cycles

Suppose I am given an element $f$ in $S_7$, so that $f=(4,1,5)(3,2)(1,5,3)$. I want to write this as a product of disjoint cycles. How would I do this?
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Probability of disjoint cycles.

Let $c_1,c_2\in S_n$ be two disjoint cycles of length $|c_1|$ and $|c_2|$ respectively. Let $I(c_i)$ be the coordinates on which permutation $c_i$ acts at $i\in\{1,2\}$. Note by choice we have ...