For questions related to permutations, which can be viewed as re-ordering a collection of objects.

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De-arrangement in permutation and combination [duplicate]

This article talks about de-arrangement in permutation combination. Funda 1: De-arrangement If $n$ distinct items are arranged in a row, then the number of ways they can be rearranged such ...
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normal subgroups of infinite symmetric group

I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the ...
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Counting in how many ways rocks can be put in boxes

How can I figure out the following questions? How many possible combinations can be done by having 26 boxes and 15 red rocks, and 15 black rocks? Each box can have up to 15 rocks in it. We can have ...
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Finding an imprimitive group on $12$ letters

By definition a permutation group $G$ acting on a set $\Omega$ is called primitive if $G$ acts transitively on $\Omega$ and $G$ preserves no nontrivial blocks of $\Omega$. Otherwise, if the group does ...
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Fixed Block is an orbit?

Reviewing some of my old questions here, I am stuck at a comment in which Prof. Holt gave me an interesting example (A small one) about non-transitive $1/2-$transitive group. Here is the link ...
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103 views

Need help about $P\Gamma L_2(q)$, $q=4,3$

I am asking kindly, For which values of $n$ we have $$S_n≅P\Gamma L_2(3),S_n≅P\Gamma L_2(4)$$ This may be correct if we replace $S_n$ by $A_n$. Any help will be appreciated. :) Edit (JL): ...
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Is $P_\omega$ a $p$-Sylow subgroup of $G_\omega$

We have the following theorem Let $G$ be a group, acting on a set $\Omega$ and let $p^m\Bigm||\omega^G|$ wherein $p$ is prime and $\omega \in \Omega$. If $P$ is a $p$-Sylow subgroup of $G$ then, ...
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About the stablizer of an element, $G_\alpha$

After reading some notes about permutation groups, I have tackled with this really simple question I hope it is not a ridiculous question. :) From the first chapters of any book about these kinds ...
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Simple permutation/combination question

In how many ways I can arrange six books on different subjects in a row such that the Math book is always to the left of history book (not necessarily adjacent) ?
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609 views

Identical balls arrangement in a circle

Six identical yellow balls and four identical red balls are to be arranged in the circumference of a circle. In how many ways it can be done?
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121 views

Polynomials and partitions

There is a question I have based on the fact: If you take a quadratic polynomial with integer coefficients, and take the set (1,2,3,4,5,6,7,8), and make a partition A=(1,4,6,7), and B=(2,3,5,8), and ...
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92 views

Finding a subgroup of $PSL_2(11)$

Here is my problem: Let $X=\begin{pmatrix} 10 & 8 \\ 8 & 1 \end{pmatrix}$ and $Y=\begin{pmatrix} 5 & 7 \\ 5 & 5\end{pmatrix}$ be two elements of $SL_2(11)$. Find a subgroup of ...
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Asking about $M(q^2)$ and its order

I am doing some handy calculation for showing that $M(q^2)$ is a group acting on set $\Omega =GF(q^2)∪\{\infty\}$ $3-$transitively wherein $q^2$ is odd in the way Dennis Gulko showed. So I need the ...
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1answer
86 views

Are there any conditions for $G$ until above action has non-trivial kernel?

Let $G$ is a group and $H$ be a subgroup of it. Then $G$ can act on the following set $$\Omega= \{Hg|g\in G\}$$ by $\forall Hg\in\Omega$ and $x\in G$; $(Hg)^x=Hgx$ (I don't know if I can call this ...
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Finding a counter Example

I am reading the following theorem: Let $G$ is a group acting on a set $\Omega$ transitively and let $B\neq\emptyset $ be a block of $G$. Then $|B|$ divides $|\Omega|$. From the first step till ...
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116 views

Does this equation imply the Class Equation?

At this time, I am reading the following theorem. Let $G$ be a group acting transitively on a set $\Omega$. Then \begin{equation} |G|=\sum_{g\in G}\chi(g) \tag{$\clubsuit$} \end{equation} ...
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423 views

Order of Perfect shuffle of 2n cards

How many times do we need to perfectly shuffle a deck of 2n cards for them to return to their original order. for $n=3$ we have the permutation $(142)(356)$ I am interested when 2n+1 is prime. Here ...
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$PGL_2(q)$ acts on $\Omega$ $3-$transitively?

Anyone who studies Permutation Groups will be encountering the following definition: A group $G$ acting on a set $\Omega$ is said to be “Sharply m-Transitive” iff $$\forall (a_1,a_2…,a_m) , ...
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Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$

There is a famous Theorem telling that: For $n≥5$, $A_n$ is the only proper nontrivial normal subgroup of $S_n$. For the proof, we firstly start with assuming a subgroup of $S_n$ which ...
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139 views

How to find a canonical member of an equivalence class of matrices under row and column swaps?

Call two matrices "swap-equivalent" if one matrix can be transformed into the other via some sequence of row swaps and column swaps. I'd like a computationally efficient algorithm that can transform ...
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85 views

What is the length of a maximal deranged sequence of permutations

We were playing a home-made scribblish and were trying to figure out how to exchange papers. During each round, you'll trade k times and each time you need to give your current paper to someone who ...
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Similarity between two nPn permutations of the same set.

Given two $nPn$ permutations of the same $n$-sized set, how can one find out the similarity between these permutations over the interval $[0, 1]$?
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Solving this permutation

I know this is an extremely noob question, but I need some help. since I am stuck Prove the formula $$p(n,r) = \frac{(n + 1 -r) \; (r^2 - 3r + 3) \; (r-2)!}{n!}$$ from this answer.
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Permutations: Given $P^4$, how many $P^1$s are possible?

Let $P^0$ be the identity tuple $(1,2,...,N)$ Let $P^{i+1}$ be the tuple after a permutation $P$ is applied to $P^i$. For example, if $P$ is $(2,1,3,6,4,5)$ than: $$\begin{align} P^0 &= ...
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Can $G≅H$ and $G≇H$ in two different views?

Can $G≅H$ and $G≇H$ in two different views? We have two isomorphic groups $G$ and $H$, so $G≅H$ as groups and suppose that they act on a same finite set, say $\Omega$. Can we see $G≇H$ as permutation ...
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Does this explanation of derangements on Wikipedia make sense?

On the Wikipedia page on derangements, the following description is given about how to count derangements: Suppose that there are $n$ persons numbered $1,2,\ldots,n$. Let there be $n$ hats also ...
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How to know if its permutation or combination?

I have a question, In how many ways can 6 tosses of a coin yield 2 heads and 4 tails? Now, to me the question clearly seems to be of permutation as they have ...
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Primitivity implies transitivity?

I am noting a simple problem about a permutation group from "Permutation Group" By J.Dixon, its answer and my attempt to understand it in details: Q: A primitive permutation group $G(≠1$) is ...
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Permuting 15 books about 2 shelves, with at least one book on each shelf.

From Descrete and Combinatorial Mathematics: An Applied Introduction: Pamela has 15 different books. In how many ways can she place her books on two selves so that there is at least one book on ...
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Nearest matrix in doubly stochastic matrix set

Suppose $\mathcal{D}_N$ denote an $N\times N$ doubly stochastic matrix, given any element $M\in \mathcal{D}_N$ , the singular value decomposition for $M$ is $$ M=USV'$$ where $U$ and $V$ are two ...
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Is there any permutation $x≠1$ leaving at least $n-2k$ letters fixed at this group?

This question has an answer which I am noting both here. Q: Suppose that $G$ is permutation group of degree $n$. If for an integer $k$ where $4≤2k≤n$ we have $|G|≥(n-k)!k$ then $G$ contains a ...
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unique permutations

Let $X$ be a set of permutations with repetitions of numbers from $1$ to $n$ Let $Y \subseteq X$ be unique if for all $\sigma, \pi \in Y$, $1 \leqslant i < j \leqslant n$ the fact that $\pi(i) = ...
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If $G$ is a transitive permutation group then $\mathrm{fix}(G_\alpha)$ is a block

I am new here and don't know much about Latex so, I attach my question from Permutation Groups by J. Dixon. I hope to get a help for it: 1.6.5 Let $G$ be a transitive subgroup of ...
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Finding what $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ is isomorphic to

I am doing an exercise that asks me to find what $\langle(135)(246),(12)(34)(56)\rangle\subset S_{6}$ is isomorphic to. I am allowed to only use the groups $D_n,S_n,\mathbb{Z}_n$ and the direct sums ...
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Need help in determining where this pascal's triangle-like sequence comes from.

I have a very interesting problem in that a program that I am running has generated a sequence of numbers that act like the pascal's triangle but have somehow built more structure into it. I have been ...
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how many 5-digit numbers satisfy the following conditions

How many five-digit numbers divisible by 11 have the sum of their digits equal to 30? I am able to get the 5-digit numbers divisible by 11 and I am also able to get the five-digit numbers whose sum ...
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Counting question on permutation matrices with rotation and imprinting

Please read question of distinct permutation matrices with rotation at first, then new counting questions are below: For a distinct $N\times N$ zero-symmetry permutation matrix, we could rotate it 3 ...
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Teaching permutations, How to?

I posed this question to my niece while teaching her permutations: Given four balls of different colours, and four place holders to put those balls, in how many ways can you arrange these four ...
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Combinatorics Issue without repetitive combinations

We have 26 Boxes Labeled: Box 1, Box 2, Box 3 and so on. The boxes are in a specific order. We also have 15 rocks. Rocks are all identical. meaning Rock 1 is no different then Rock 2, or does not have ...
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If $G\subseteq S_n$ is a subgroup acting transitively on $\{1,\ldots,n\}$, then a nontrivial normal subgroup $N\subseteq G$ has no fixed points

Let $G$ be a subgroup of $S_n$, which acts transitively on $I= \{1, \ldots, n \}$. Let $N$ be a nontrivial normal subgroup of $G$. Then $N$ has no fixed points in $I$.
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Is there a fast way to determine the conjugacy classes of $S_5$?

I was working towards proving $A_5$ is the only nontrivial normal subgroup of $S_5$. To do this, I wanted to find a set of representatives of conjugacy classes of $S_5$, and their respective orders. ...
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213 views

Any permutation in symmetric group n can be rewritten as a composition of transpositions

I just want to show that a permutation can be written as a composition of transpositions. I cannot use cycles.
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How many non-isomorphic permutation selections are on an arbitrary N x N square matrix with rotations applied?

My question is an extension to a classic one: On a square $N \times N$ grid, select exact $N$ cells that satisfy condition: only one cell selected in same row and column. How many solutions will ...
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301 views

Number of permutations with a certain number of fixpoints

Given a set of $n$ mutually distinct elements, how many permutations are there such that exactly $k$ of the permuted elements stay at the same place? Example Let's take the set $\{A,B,C,D\}$. The ...
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Why is the holomorph of a group $G$ a group of transformations of $G$?

I'm reading a definition of the holomorph of a group $G$, where it is defined as $G_L\operatorname{Aut}(G)$, where $G_L$ is the group of left translations of $G$, that is, the maps of form ...
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How to find the last non-zero digit in ${^n\!P_k} $?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
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What will be total number of solutions of $a+b+c = n$?

Please tell me how to find the total number of intergral solutions of $$ a+b+c=n $$ I already know that total number of solutions will be $(n+3-1)c(3-1)$. But what will be the case when a varies from ...
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$i$ balls to paint $k$ colors, and exactly $k' < k$ colors should be used. How many ways to paint?

Another ball-painting problem: assume that we have $i$ balls (with numbered labels, so order is sensitive), and $k$ different colors. Now we need to paint these balls using these colors so that ...
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The sgn function and permutations

Let $P=\{(i, j)|1\leq i<j\leq n\}. $For $\sigma\in S_n$, define $\operatorname{sgn}\colon S_{n}\rightarrow \{\pm 1\}$ by $$ \operatorname{sgn}(\sigma)=\prod_{(i, j)\in ...
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Can two finite sequences be considered permutations if their products and sums are equal?

Given two finite ordered sequences with possibly non-unique elements all greater than one: $A,B \in \mathcal{Z}_{>1}$. Given that we have: \begin{eqnarray} |A| &= |B| \\ \Pi_{x \in ...