A permutation group is a group consisting of permutations of some given set $M$.

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Is a primitive permutation group, indecomposable?

Is a primitive permutation group, indecomposable?
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$G=\{f_n(x):n\in \mathbb{Z}\}$ is cyclic

Define $f_n(x)=x+n \;\;\forall n \in \mathbb{Z}$. Let $G=\{f_n:n\in \mathbb{Z}\}$. I proved that $G$ is a subgroup of $S_\mathbb{R}$ ($f_n$ is a permutation of $\mathbb{R}$), and now I am trying to ...
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Has this partition a name?

The decomposition in disjoint cycles in $S_n$ was inspiring for this question. I found out (if I did nothing wrong) that more generally a bijection $f:X\rightarrow X$ induces a partition of $X$ ...
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Series of multi sets

Given that a set of numbers $K = \{n_1, n_2, n_3, ... \}$. Multiple subsets are formed by randomly extracted numbers from $K$. Then series are formed by extracting numbers from the subset orderly. ...
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Alternating group $A_n$ generated by its subgroups $A^{(i)}_{n-1}$

Suppose $A^{(i)}_{n-1}$ is the alternating group taking the $n-1$ numbers $\{ 1, 2, \cdots, i-1, i+1, \cdots, n \}$ that are the domain of even permutations in it, where $n \geq 4$ and $i =1, 2, ...
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Something that is true for every element of $\text{Sym}(\Bbb{N})$

I'm trying to prove that: Every element of $\text{Sym}(\Bbb{N})$ can be written as $f\circ g$ for some $f,g\in \text{Sym}(\Bbb{N})$ with $f^2=g^2$. But I can't even prove this for ...
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Uniqueness of the direct product decomposition of inclusions of finite groups

This post is a generalization of Uniqueness of the direct product decomposition of finite groups. Here we look inclusions of finite groups $(H \subset G)$ instead of just finite groups. Definition: ...
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In $S_4$, what does the expression for the cyclic group $\langle (13),(1234)\rangle$ mean?

In $S_4$, what does the expression for the cyclic group $\langle (13),(1234)\rangle$ mean? I apologize if this is too basic, but I haven't come across such an expression anywhere in my book. Also, ...
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Generators of the Symmetric group $S_3$

I am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not found ...
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Prove that $\operatorname{Aut}(S_n)$ isomorphic to $S_n$ when $n\geq 3$ and $n \neq6$.

Prove that $\operatorname{Aut}(S_n)$ isomorphic to $S_n$ when $n\geq 3$, and $n \neq 6$. I can see that the automorphisms of $S_n$ have the same structure as $S_n$. But I am having trouble ...
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on primitive group actions with abelian stabilizers

I am trying to solve the following exercise from Dixon and Mortimer: Let $G$ be a finite primitive permutation group with abelian point stabilizers. Show that $G$ has a regular normal elementary ...
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A Gap code for the alternating group $A_4$

I need a GAP code for checking the following question: Is it true that for every subset $A$ of the alternating group $A_4$ with $4$ elements there exists a subset $B$ of order $3$ such that ...
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Number of elements of order $2$ in $S_n$

How many elements of order $2$ are there in $S_n$? Using combinatorics I arrived at this: For $n$ even ($n=2k$) there are ${n\choose2}+{n\choose 2}{n-2\choose 2}\dfrac{1}{2!}+{n\choose 2} ...
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How to “see at a glance” the solution to the exercise “Show that $\langle (1,2,3… n),(1,2,3… m)\rangle$ contains a 3 cycle, if $1 < n < m$”?

I tried the commutator of the generators and it worked, but I had no real justification for making that computation. Is there a perspective from which trying the commutator is the "obvious" thing to ...
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Maximal height of subgroups in $S_n$?

In the process of solving some exercise, I became curious about the maximum height of a chain of subgroups in $S_n$. More specifically - what is the maximum length k of a chain of subgroups $\{e\} ...
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A question on the automorphism of simple graph with distinct eigenvalues of adjacency matrix

Let G be a graph. If G is simple(i.e no loops), and the eigenvalues of adjacency matrix A are distinct, then the automorphism of G is abelian. It seems that the automorphism from G to G itself is only ...
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Regarding the composition of permutations…

So I have the permutations: $$\pi=\left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 2 & 3 & 7 & 1 & 6 & 5 & 4 & 9 & 8 ...
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$A_n \times \mathbb{Z} /2 \mathbb{Z} \ \ncong \ S_n$ for $n \geq 3$

I try to show that $A_n \times \mathbb{Z} /2 \mathbb{Z} \ \ncong \ S_n$ for $n \geq 3$. It is not hard to show the statement for $n=3$. We have $$ A_3 \times \mathbb{Z} /2 \mathbb{Z} \ \cong \ ...
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Show that there is just one subgroup $H \subset S_4$ such that $[S_4:H] = 2$

I had to show that $S_4$ has one subgroup of index $2$. Below, you'll find what I tried to do so Of course, $A_4$ is a subgroup index $2$. To show that there is not another subgroup with the same ...
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The permutation $\lambda_g \quad : \quad G \longmapsto G \quad : \quad x \mapsto gx$

Let $G$ be a group containing $2k$ lements where $k$ is odd. Let $g \in G$ of order $2$ and define $$ \lambda_g \quad : \quad G \longmapsto G \quad : \quad x \mapsto gx $$ I had to show that ...
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Show if $\pi,\sigma$ are any permutation s.t $(\pi\sigma)^2=\pi^2\sigma^2$, then $\pi\sigma=\sigma\pi$…

As the title says, the problem is: Show if $\pi,\sigma$ are any permutation s.t $(\pi\sigma)^2=\pi^2\sigma^2$, then $\pi\sigma=\sigma\pi$. There is a theorem that states If $\pi,\sigma$ are ...
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Calculating sign of a permutation

Here's the permutation: $\pi\sigma= \left( \begin{array}{ccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 5 & 6 & 2 & 9 & 1 & 7 & 8 & 3 & 4 ...
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To calculate number of combination of sequences having 1 and 2 alternating sequences of R and S.

I have a sequence of 6 letters containing 2 P, 2 R , 1 Q and 1 S. I have PPQ, now I have to add two R and one S in that, these can be placed anywhere. There will be total 60 possible ways to do that ...
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pemutation representation that confuses me a lot recently

For any group G, define group action on a set A. There will be a permutation representation of that group action. I am kind of confused why the permutation representation can be used to reflect the ...
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When is a power of $m$-cycle is also an $m$-cycle?

I have a question taken from Abstract Algebra by Dummit and Foote ($pg.33$, $q.11$): Let $\sigma\in S_{n}$ be an $m$-cycle. Show that $\sigma^{k}$ is also an $m$-cycle iff $\gcd(k,m)=1$. My ...
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I want to decompose a tensor product using Littlewood-Richardson rule, How do I find the component of this in each irreducible space?

Let me set up the notation I am using. $(abc,de)$ denotes the standard Young tableau where the first row is $abc$ and the second row is $de$. Each young tableau corresponds to the young symmetriser, ...
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Combination of $4$ digit numbers not divisible by $5$

The number of $4$ digit numbers which are not divisible by $5$ that can be formed using the digits $(0,2,4,5)$ if digits are not repeated is?
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For a group $G$ acting transitively, is $G_{\alpha}$ contained in stabilizer of some block $\Delta$ if $\alpha \in \Delta$?

This question refers to permutation groups, in particular the primitive ones, and block systems. Let $G$ be a finite group acting on a set $\Omega$, and consider some partition $\Delta_1 \cup \ldots ...
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Quaternion Group as Permutation Group

I was recently, for the sake of it, trying to represent Q8, the group of quaternions, as a permutation group. I couldn't figure out how to do it. So I googled to see if somebody else had put the ...
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All permutation matrices that convert one Hadamard matrix into another Hadamard matrix.

Given a Hadamard matrix $H$, I know that applying row and column permutations, along with multiplying a row or a column with a -1 results in another Hadamard matrix $H^{'}$ equivalent to the first. ...
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Elements whose orders are multiple of $p$ [closed]

Let $G$ be a non-solvable group, $N$ an abelian minimal normal $p$-subgroup of order $p^r$ with $p\notin \pi(G/N)$, $N=C_G(N)$ and $K=G/N\cong A_5$. By these assumption we can conclude that $G$ has ...
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Computing values of centralizers in a non-solvable group with a given property

A finite group G satisfies property $P_n$ if for every prime integer $p$, $G$ has at most $(n−1)$ non-central conjugacy classes the order of the representative element of which is a multiple of $p$. ...
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multiplication of permutation

I didn't find any good explanation how to perform multiplication on permutation group written in cyclic notation. For example, $a=(1\ 3\ 5\ 2)$, $b=(2\ 5\ 6)$, $c=(1\ 6\ 3\ 4)$, $ab=(1\ 3\ 5\ 6)$, ...
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For $k$ random perms of an $n$-set $\mathrm{Pr}[\sigma_1\cdots\sigma_k=\sigma_k\cdots\sigma_1]\xrightarrow{k\rightarrow\infty}\frac{2}{n!}$?

Q. Fix $n \geq 2$, and choose $k$ random permutations $\sigma_1\sigma_2\cdots\sigma_k \in S_n$ uniformly at random. Is true that ...
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Probability of $\alpha\beta\gamma=\gamma\beta\alpha$ for random permutations of a finite set?

Following up on my previous question Probability that two random permutations of an $n$-set commute?, here's a related question for three elements. Q: If $\alpha,\beta,\gamma$ are chosen uniformly ...
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Probability that two random permutations of an $n$-set commute?

From this MathOverflow question: It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. -- Benjamin ...
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Commutative permutations

Observation: Two disjoint permutations are commutative. For e.g $\ (1357)\in S_8$ and $\ (2468)\in S_8$. Formulate and prove(if possible) a Theorem generalizing your observation about commutative ...
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Abstract Algebra:Permutations

1)Express the permutations $\alpha=(24)\in S_4$ and $\beta=(1)\in S_5$, as sets. a) Describe the permutations which are reflexive b) What types of permutations are partial orders. Attempt of a ...
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Permutations: If I know $\alpha$ and the cycle structure of $\alpha\beta$, can I find $\gamma$ for which $\gamma\beta$ also has this cycle structure?

Suppose we have two permutations $\alpha$ and $\beta$ (of a set $S$ of size $|S|=n$), and I know $\alpha$ and the cycle structure of $\alpha\beta$. But I don't know $\beta$. Can I find a ...
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identifying a subgroup of $S_8$ generated by 4-cycles

Let $G \subseteq S_8$ be the subgroup generated by some 4-cycles. If we number the elements $1,2,\dots, 8$, the 4-cycles are $(1234),(5678),(1485),(2376),(1265),(4378)$ I am not sure if I have ...
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Infinite set and Permutation group

Suppose $X$ a infinite set and $S_X$ is the permutation group of $X$. Prove that any proper subgroup of $S_X$ has infinite index.
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How do i find the order of a permutation in the group $S_n$

How can i define the order of a permutation without doing the permutation again and again? Example: say $σ=(1-->2,2-->3,3-->5,4-->1,5-->4)$ in $S_5$.
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Blocks in permutation group theory (D&F)

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text: Let $G$ be a transitive permutation group on the finite set $A$. $A$ block is a nonempty subset $B$ of $A$ ...
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Relation between permutations and fourier transform?

i dont know if this is already addressed somewhere (searching around did not find sth). The motivation is to find a way to generate or produce permutations using concepts from continuous mathematics ...
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Are all Sylow 2-subgroups in $S_4$ isomorphic to $D_4$?

I was assigned to show that every Sylow 2-subgroups in $S_4$ is isomorphic to $D_4$. So I figured, since $|S_4|=24=2^3\cdot 3$, every Sylow 2-subgroup has either the form: $\langle ...
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Finding the centralizer of a permutation

I need to find the centralizer of the permutation $\sigma=(1 2 3 ... n)\in S_n$. I know that: $C_{S_n}(\sigma)=\left\{\tau \in S_n|\text{ } \tau\sigma\tau^{-1}=\sigma\right\}$ In other words, that the ...
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Example of non-Abelianness of symmetric group for graphs

I know that for $n \ge 3$, $S_n$ is non-Abelian. I would like to work out an example in terms of graphs so to make it sure that I understand it right. A symmetric group of graphs of four vertices, ...
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Two 3-cycles generate $A_5$

I want to solve the following exercise, from Dummit & Foote's Abstract Algebra Let $x$ and $y$ be distinct 3-cycles in $S_5$ with $x \neq y^{-1} $. Prove that if $x$ and $y$ do not fix a ...
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Largest symmetric group contained in alternating group

I know that for $n \geq 3$, the alternating group $A_n$ contains a subgroup which is isomorphic to $S_{n-2}$, namely $$\langle \{(i \;i+1)(n-1 \;n):1 \leq i \leq n-3\} \rangle.$$ I was wondering what ...
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Do cyclic permutations of rows and column entries generate all permutations?

Background: I am interested in the group of permutations of the entries of a general $m\times n$ matrix. In particular, I am interested in (1) interesting sets of simple generators for this group ...