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

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2answers
103 views

Help on groups and symmetries homomorphisms

I need help on question 1 I am preparing for my test: 1) Find a homomorphism $\alpha: A_4 \to\mathbb Z_6$ such that $\ker (\alpha) = K$ where $K$ is the normal subgroup $K= \{(1), (12)(34), ...
4
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2answers
391 views

How many 4 worded sentences can a list of 5 words make if two of them must be in that sentence?

Suppose we have: I am new at this - (5 words) how many 4 worded sentences can we make with this if "new" and "this" must appear in the sentence. I think its : .# of sentences we can make with any ...
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1answer
105 views

Finding the number of distinct elements in a group generated by a permutation and a function over tuple.

Okay, I'm not sure if the title is correct but is there any general way of finding the number of distinct elements that are generated by some function (for example $*$) and a permutation (in this case ...
3
votes
1answer
356 views

Permutations of a set with a conditional subset

Using the digits 1, 2, 3, 5, 6, 8, 0 only once, how many 4-digit numbers could be constructed if the number is even? This is an exercise from an online course I'm taking. The given solution suggests ...
6
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3answers
5k views

Distinct permutations of the word “toffee”

What does distinct permutations mean and how many distinct permutations can be formed from all the letters of word TOFFEE?
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0answers
297 views

sum with permutations

Let $a$ be vector in $R^{2m}$. And let $S_{2m}$ be group of all permutations on the set $\{1,\dots,2m\}$. I would like to calculate $$ \sup_{\pi\in S_{2m}}\sum_{d(\sigma, ...
0
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1answer
92 views

permutation within a combination

I am having a bit of a problem with the following question: There are 16 balls, 5 red, 8 blue, and 3 green. The result is the ordered list of the first 4 balls only (although all balls have been ...
8
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1answer
183 views

Subgroups between $S_n$ and $S_{n+1}$

Lets look at $S_n$ as subgroup of $S_{n+1}$. How many subgroups $H$, $S_{n} \subseteq H \subseteq S_{n+1}$ there are ?
4
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1answer
280 views

question involving Markov chain

Let $S_{2m}$ be the group of all permutations $\pi$ of $\{1, 2, \ldots, 2m\}$. The following transition kernel $S$ generates the random transposition walk $$ Ch(\pi, \pi')= \begin{cases} \frac{1}{2m} ...
2
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2answers
249 views

Permutation problem scheduling games

There are: 14 teams $t$ numbered from 0 to 13 13 different games $g$ numbered from 0 to 12 13 rounds $r$ numbered from 0 to 12 I want to make a planning such that: each team plays against ...
2
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2answers
53 views

Combinatorics: how many unique albums…

I want to record a set of music albums so that each one is unique. Each album has ten tracks, and I've recorded 4 versions of each track. How many unique albums can I compile so that no two albums has ...
2
votes
2answers
96 views

Bijection from $S_{n-1}$ to $\{\sigma \in S_{n} : \sigma(k) = j \}$

Let $n$ be a natural number. Let $k$ be an element of $\{1, \ldots , n\}$. For each j in $\{1, \ldots , n\}$, I want to find a bijection $f_j$ from $S_{n-1}$ to $\{\sigma \in S_n : \sigma(k) = j ...
0
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2answers
81 views

About $|\operatorname{Sym}(\Omega)|$ when $\Omega$ is an infinite set.

Here is a problem: Show that if $\Omega$ is an infinite set, then $|\operatorname{Sym}(\Omega)|=2^{|\Omega|}$. I have worked on a problem related to a group that is $S=\bigcup_{n=1}^{\infty } ...
2
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6answers
110 views

Permutation seating arrangment 5!.3!

What wold be the answer for this How many ways can 3 boys and 4 girls sit in a row if all the boys are sit together. Answer listed as $5!\cdot3$! $4+3 = 7$ what is 5 doing here? someone please ...
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2answers
2k views

How to solve this permutation math problem?

In how many ways can 4 girls and 2 boys sit at a movie theater row with 6 seats if a girl must be seated at each end.
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4answers
189 views

Selection using permutation and combination

From 4 men and 4 ladies a committee of 5 is to be formed. The committee consists of a president, vice president and three secretaries. What will be the number of ways of selecting the ...
1
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4answers
5k views

Difference between permutation and combination?

Permutation: $$P(n,r) = \frac{n!}{(n-r)!}$$ Combination: $$C(n,r) = \frac{n!}{(n-r)!r!}$$ Apparently, you use combination when the order doesn't matter. Great. I see how a combination will give you ...
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2answers
2k views

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 ...
9
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2answers
512 views

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 ...
3
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1answer
161 views

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 ...
1
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1answer
65 views

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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1answer
128 views

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 ...
3
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1answer
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): ...
2
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0answers
74 views

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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2answers
90 views

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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4answers
511 views

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) ?
3
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2answers
549 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?
3
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1answer
119 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 ...
0
votes
1answer
90 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 ...
1
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1answer
55 views

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 ...
0
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1answer
80 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 ...
1
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1answer
129 views

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 ...
2
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1answer
110 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} ...
0
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1answer
350 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 ...
3
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2answers
77 views

$PGL_2(q)$ 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) , ...
4
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1answer
578 views

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 ...
2
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1answer
135 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 ...
3
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0answers
152 views

Organizing Wikipedia's list of permutation topics [closed]

What would be a sensible way of organizing Wikipedia's List of permutation topics into sections and possibly sub-sections, with similar topics grouped together in a section? Later edit: Per Jack ...
3
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1answer
82 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 ...
1
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2answers
305 views

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 $[0, 1]$?
1
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1answer
83 views

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.
8
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1answer
170 views

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 &= ...
22
votes
2answers
606 views

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 ...
3
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1answer
101 views

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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5answers
9k views

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 ...
1
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1answer
175 views

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 ...
3
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1answer
2k views

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 ...
4
votes
2answers
295 views

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 ...
1
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1answer
73 views

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 ...
1
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2answers
161 views

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) = ...