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

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Hausdorff quotient space

consider a smooth manifold $M$ and a group action, i.e. a group homomorphism $\phi: G\rightarrow S(M)$, where $S(M)$ denotes the group of diffeomorphisms of $M$. Suppose that for all $K\subset M$ ...
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23 views

Proof involving the group of permutations of $\{1,2,3,4\}$.

Let $\sigma_4$ denote the group of permutations of $\{1,2,3,4\}$ and consider the following elements in $\sigma_4$: ...
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1answer
17 views

What does it mean a transitive permutation?

let $X$ be a finite set. Let G be a group. What is the meaning of $G$ is a transitive permutation on set $X$?
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1answer
546 views

$N^\text{th}$ (in lexicographical order) term of balanced brackets string

We have the following balanced brackets permutations of length $4\cdot2$ in lexicographical order: ...
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2answers
24 views

What is the probability that nobody receives the same ranking twice?

Four players compete in a tournament and are ranked $1$ to $4$. They then compete in another tournament and are again ranked from $1$ to $4$. Suppose that their performances in the second ...
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0answers
32 views

Permutations of $S_n$ whose order divides a positive integer $m$

For which $n,m \in \mathbb{N}$ is $$K_m = \{\sigma \in S_{n}: \text{ord}(\sigma) \text{ divides } m \}$$ a subgroup of $S_{n}$? How does one approach such a problem?
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1answer
30 views

Order of a subgroup generated by two permutations

Let $$\alpha=(1,3,12,7)(8,5,6,2,11)(4,9,10)$$ $$\beta = (1,5)(6,8,11)(12,3,2,7)(4,9,10).$$ How does one prove that a subgroup G that contains $\alpha$ and $\beta$ has order $o \ge 120$? ...
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1answer
45 views

For which $p$ is $G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$ a subgroup of $S_{12}$?

Let $p$ be a prime number. I've to figure out for which $p$ $$G_p = \{\sigma \in S_{12}: \text{ord}(\sigma) \text{ divides } p \}$$ is a subgroup of $S_{12}$. I realize that we have to ...
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0answers
21 views

License Plate Permutations

A state has changed its license plate numbering system for the three largest counties. Before the change, each plate had the number 1, 2, or 3, followed by either one or two letters, followed by 3 ...
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2answers
34 views
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0answers
15 views

Disjoint cycle Decomposition.

Let $\sigma =(a_1...a_6)$ be a $6$-cycle. Write the disjoint cycle decomposition of $\sigma^2$ and $\sigma^3$. I know that $a_1*a_1=\epsilon $ but does this mean that $\sigma^2=\epsilon$ and then ...
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0answers
24 views

Looking for mathematical/combinatorial and computational explanation regarding adding values in a $5 \times 4$ (matrix?) with a constraint.

Given the following matrix (not sure if I should call it that): Matrix $5 \times 4$ I want to add all possible combinations of values such that each Horse gets but one value from each Bookie. What I ...
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2answers
20 views

Why use C(n,r) instead of P(n,r) when considering how many strings can be formed in which a specific letter appears before another specific letter?

I am dealing with a problem in which I must determine how many strings can be formed by ordering the letters ABCDE subject to the conditions given. The condition that I am given is that A appears ...
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0answers
19 views

permutations and representations , sign function.

Show that the sign representation of $S_n$ is indeed a representation. attempt: Recall the sign function of a permutation is given by $\mathrm{sgn}(\pi) = (-1)^k$. Then recall a representation is a ...
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0answers
72 views

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? [on hold]

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? How should I approach this problem? Okay I think it's C(10, 2) because I already have ...
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0answers
24 views

Centralizer of $(1,2,3)(4,5,6,7,8) \in A_{11}$

Let $$\tau = (1,2,3)(4,5,6,7,8) \in A_{11},$$ where $A_{11}$ is the group of even permutations. Let $H$ be the centralizer of $\tau$. I can easily prove that $H$ is a subgroup of $A_{11}$. How do I ...
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3answers
78 views

Understanding the definition of the sign of a permutation , $\operatorname{sgn}(\pi) = (-1)^k$ .

I am trying to understand the definition of the sign of a permutation $\pi$. My textbook only mentions that $\operatorname{sgn}(\pi) = (-1)^k$ , where $k$ is the number of transpositions . But I ...
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1answer
11 views

Finding the smallest exponent $k$ for a non-cyclic permutation $\sigma$, so that $\sigma^k = id$.

What I am aware of (1) A cyclic permutation is a permutation that consists of a single nontrivial cycle (cycle of length $> 1$). Let $k$ be the length of the cyclic permutation $\tau$. Therefore ...
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1answer
62 views

Extend isometry on some cube vertices to the entire cube

Let $K\subset V=\{-1,1\}^n$ be a set of vertices of the $n$-dimensional hypercube $D=[-1,+1]^n$ and let $f:K\to V$ be an isometry with respect to the Euclidean metric inherited from $\mathbb R^n.$ We ...
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1answer
21 views

Summation of series with binomial coefficients

The value of $$\sum {n\choose n-r} (n-r) \sin(r\cdot \pi/n)$$ where $r\in (0 ..,n)$ is equal to? I think the question can be solved by writing the series in reverse order but I am not able to solve ...
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1answer
42 views

Sets and probability

The number of total outcomes of an experiment are $25$. If $A$ and $B$ are two non-empty independent events of the experiment such that outcomes in favour of event $A$ are $15$, then the minimum ...
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1answer
57 views

Proper subgroup of $S_{15}$ that strictly contains $\sigma $

How does one prove that: There exists no cyclic proper subgroup of $S_{15}$ that strictly contains the following permutation (of order $10$)? $$\sigma = (1, 2,3, 4,5, 6, ...
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8 views

Why are two transitive actions of a group G equivalent if there exist an automorphism of G swapping two point stabilisers?

Let $G$ be a group acting transitively on two sets $\Omega_{1}$ and $\Omega_{2}$. Also let $w_{i}\in\Omega_{i}$ and suppose there exists $\alpha\in Aut(G)$ such that $\alpha(G_{w_{1}})=G_{w_{2}}$, ...
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2answers
29 views

Divisors of $75600$ of the type of $4n+2$

Find the total no. of divisors of $75600$ which of the type of $4n+2$ where $n\in \mathbb{N}$ and $75600=2^4 \cdot 3^3 \cdot 5^2 \cdot 7^1$ Now I think divisors of type $(4n+2)$ should be of ...
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1answer
30 views

Number of permutations with given cyclic structure

If $\sigma$ is a permutation made up by the disjoint cycles $\tau_1, \dots, \tau_r$ (including those of length $1$), we call structure of $\sigma$ $$(l_1, \dots, l_r),$$ where $l_1, \dots, l_r$ are ...
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1answer
47 views

Prove that the set $U = \{(123), (124), … , (12n)\}$ can be used to generate $A_n$.

A hint is provided with the proof prompt: $(abc) = (1ca)(1ab)$, $(1ab) = (1b2)(12a)(12b)$, and $(1b2) = (12b)^2$. My idea: $(1ab) = (12b)(12b)(12a)(12b)$. To solve for the other half of $(abc)$, I'm ...
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1answer
27 views

If $\sigma$ is a product of $k$ transpositions, prove that $k \equiv$ inv $(\sigma)$ (mod $2$).

An inversion in a permutation $\sigma = \sigma_1...\sigma_n$ is a pair of indices $i < j$ such that $\sigma_i > \sigma_j$. Let inv($\sigma$) be the number of inversions of $\sigma$. If ...
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4answers
36 views

Number of arrangements in which no two persons sit side by side

There are $10$ seats in the first row of the theater out of which $4$ are to be occupied. Find the number of ways of arranging $4$ people so that no two people sit side by side. Making different ...
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2answers
12 views

Possible orderings when the items are not unique?

First of all, I'm sure this question has been answered somewhere on the web, but I am just starting probability and I don't have the vocabulary to know what to look for, which is why I am asking here. ...
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0answers
32 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
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4answers
618 views

Are there any Symmetric Groups that are cyclic?

Are there any Symmetric Groups that are cyclic? Because I have been doing some problems and I tend to notice that the problems I do that involve the symmetric group are not cyclic meaning they do not ...
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1answer
55 views

Subgroups of $G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}$ [on hold]

Let $$G = \{\sigma \in S_{14}: \mathrm{ord}(\sigma)\mid 12 \}.$$ Is this a subgroup? By examining particular examples, one can see that it is not, since it is not closed under composition. However, ...
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1answer
48 views

Order and generators of intersection of cyclic groups

Let $\sigma$, $\tau$ be two permutations of $S_n$. We know that $\langle \sigma \rangle \cap \langle \tau \rangle$ is a cyclic subgroup and (from Lagrange's Theorem) that its order divides ...
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15 views

All equivalent moves on a rubik's cube

Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively. Let us say that I have a permutation of the stickers on the cube written as a word ...
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1answer
46 views

How many DFA's exist with two states over the input alphabet $\{0,1\}$?

How many DFA's exist with two states over the input alphabet $\{0,1\}$? My attempt : Input set is given. So, we have 3 parts of DFA which we can change: Start state Transition Function Final ...
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29 views

Let $X : S_3 → GL_2(\mathbb{R})$ . Compute the six matrices {$X(\pi) : \pi \in S_3$} and show they faithfully represent $S_3$.

Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group ...
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1answer
38 views

The probability of being dealt at least 5 wanted cards

In a fictional deck of cards, there are 30 cards, 15 different ones (each card has an identical pair, so 15 pairs = 30 cards). I want to answer the question: I am dealt 10 cards. I wish to receive 5 ...
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0answers
31 views

What is the probaility that two random permutations have same order?

I am interested in the orders of random permutations. Since the law of the log of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one ...
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1answer
34 views

Subgroups of $S_{10}$ that contain $\langle \sigma = (123)(456)\rangle$

In $S_{10}$ let us consider the permutation $\sigma = (123)(456)$ and the cyclic subgroup $ G = \langle \sigma \rangle$. I can fairly easily find that $$\langle \alpha = (1,4,2,5,3,6)(7,8) \rangle ...
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1answer
44 views

Is it enough to determine if two finite groups are isomorphic if we can map the generators?

Heading: A General Inquiry about Finite Isomorphic Groups and Their Generators If I am given two finite groups whose generators I know, is it enough to show that by mapping the generators ...
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1answer
24 views

Probability of drawing $m$ of $A$ in $n$ cards given a deck of $d$ cards contain $a$ copies of $A$?

As in the title I'm trying to work out what the chances of drawing $m$ copies of a specific card in $n$ draws are given a deck size of $d$ containing $a$ copies of $A$. I've tried using permutations ...
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1answer
4k views

Can someone explain Cayley's Theorem step by step?

This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. ...
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1answer
19 views

Multi-stage Probability

I think the easiest way to explain what I'm having trouble with is to give an example question: A monkey is given 12 blocks: 3 Squares, 3 Rectangles, 3 Triangles, 3 Circles. Calculate the probability ...
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1answer
15 views

find all combination without overlapping

(*Constraint) Kinds of number should be limited in 3 (ex. {1,1,1,1} o/ {1,2,3} o / {1,1,2,2,5,5,4} x) And I want to find series of Integer. For example, if n = 4 (n is length of numbers) {2, 7, 2, ...
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2answers
31 views

Find the number of seating arrangements at a round table of three single men, two single women, and two families

Three single men, two single women and two families take their places at a round table. Each of the two families consists of two parent and one child. Find the number of possible seating arrangements. ...
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2answers
42 views

In how many ways can a committee of $6$ people be selected from $7$ men and $6$ women if it can contain at most one of persons A and B?

A committee of $6$ people will be formed with $7$ men and $6$ women. The oldest of the $7$ men is A and the oldest of the $6$ women is B. It is described that the committee can include at most one of ...
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1answer
19 views

Number of permutations of $S_n$ such that $\sigma^h(a) = \sigma^k(b)$

A basic result in combinatorics is: In $S_n$ there are $(n-d)(n-2)!$ permutations $\sigma$ such that $\sigma^k(a) = b$, if $a \neq b$; $d(n-1)!$ permutations $\sigma$ such that ...
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2answers
583 views

A committee of 12 is to be formed from nine women and eight men. In how many ways can this ..

Problem : A committee of 12 is to be formed from nine women and eight men. In how many ways can this be done if at least five women have to be included in a committee ? Solution : Case I : 5 W ...
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2answers
3k views

What is the usage of Permuation and Combination in the real life or Computer Programming world?

I have never understood the application of P & C, even in the software programming. I know, the principle behind P&C is counting. So, basically it makes the counting quicker. However, I could ...
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2answers
135 views

How many numbers of $10$ digits that have at least $5$ different digits are there?

In principle I resolved it as if the first number could be zero, to the end eliminate those that start with zero. The numbers that can use $4$ certain figures (for example, $1$, $2$, $3$ and $4$) are ...