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

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2
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1answer
19 views

What is the correct way to reference the indexes in a permutations cycle notation?

I have a permutation in cycle notation, say this one $$\underbrace{(123)}_{=:C_1}\underbrace{(45)}_{=:C_2}\underbrace{(6)}_{=:C_3}$$ with cycle $C_1$, $C_2$ and $C_3$. Later on I have to use these ...
4
votes
1answer
79 views

Express the permutation $\sigma = \left({}^1_5\,{}^2_8 \,{}^3_3\,{}^4_6\,{}^5_7\,{}^6_4\,{}^7_1\,{}^8_2\right)$ as a product of transpositions

$\newcommand{\lcm}{\operatorname{lcm}}$ I was hoping to get some feedback. Consider the permutation $$\sigma = \begin{pmatrix} 1&2&3&4&5&6&7&8\\ ...
-1
votes
1answer
29 views

Number of strings from 39 characterst with given length [closed]

Suppose that I have a Characters Set that has length of 39 total characters. The minimum string limit is 3 and the maximum limit of a string is 15. I just want to know how many possible outcomes it ...
1
vote
2answers
35 views

Is 0521 and 521 are same NUMBER?

In a question related to permutations, its been asked to find the number of NUMBERS with less than 5 digits that can be made using 0,1,2,3,4,5. Instead of NUMBERS, if it was ARRANGEMENTS I know that ...
2
votes
0answers
33 views

A specific question on three statements in a paper about fixed point bounded groups, its interpretation and its usage w.r.t. the Sylow theorems

This is a rather specific post, but I hope nevertheless someone can help me. I am refering to a specific paper, namely K. Mayaard, R. Waldecker, Transitive permutation groups where nontrivial ...
0
votes
0answers
20 views

Size of a permutation's support: Even vs. Odd

Here's an inquiry about permutation groups. For distinct n-cycles p and q in the symmetric group on n letters, count the letters moved by q * inv(p). Now determine how many (p,q)-pairs yield even ...
2
votes
2answers
95 views

Show that $H = \bigcup_{n > 0} S_n$ is not equal to Sym$(X)$ for $X = \{1,2,3,4,…\}$.

Let $X = \{1,2,3...\}$ be the set of positive natural numbers, $S_n$ the permutation group, and Sym$(X)$ the set of all bijections from X to X with operation composition. I have the following ...
0
votes
1answer
30 views

Can someone explain this proof that each transposition changes the parity of a permutation?

Looking at: http://dogschool.tripod.com/permutation.html It states: Let the following represent an ordering of the n numbers 1 to n. a1 a2 . . . x . . . y . . . an-1 a1 Each number in ...
1
vote
2answers
62 views

Permutations in which $d$ appears before $b$. [closed]

Consider all possible permutations of eight distinct elements $a, b, c, d, e, f, g, h$. In how many of them, will $d$ appear before $b$? Note that $d$ and $b$ may not necessarily be consecutive.
2
votes
1answer
33 views

Equivalence Class of Permutations

Problem I want to clarify some doubts that I'm having pertaining to the equivalence classes of permutations. In my Abstract Algebra course, I'm being asked to find the equivalence class of ...
1
vote
1answer
46 views

understanding cycles in $S_n$

I'm having some difficulty understanding this issue of cycles in the permutation group ($S_n$), under the subject of abstract algebra. The question is as follows (translated, pardon math ...
1
vote
0answers
36 views

I am having trouble determining if I should use a combination of a permutation in the following example

Three balls are drawn from a bag with 2 white and 3 black balls. There are 20 outcomes (sequences) in S (I think they are telling us this to say that order is important). What is the probability ...
2
votes
0answers
47 views

How to find a base of a permutation group?

How to find a base of the permutation group G=⟨x,y⟩≤S4 x=(1,2,3), y=(1,2,4)? I hear that base for G is a sequence B = [$b_1, ..., b_m$] ⊂ Ω such that the only element of G which stabilizes each $b_i$ ...
1
vote
2answers
59 views

How many sequences of length $2n$ can you make with 0-9 if none of the numbers appear $n$ times in a row?

I tried to find the total number of sequences and then minus of those with $n$ number of the same elements in a row. It seems extremely tedious so I'm wondering if anyone has a better approach to this ...
3
votes
1answer
50 views

Distinct relative ordering of four consecutive numbers

Find a permutation of the set $\{1,2,\dots, 24\}$ with the property that each consecutive group of four integers (including around the corners) has a distinct relative order. When we are looking at ...
3
votes
3answers
61 views

Can somebody explain Cayley's theorem to me?

I need some help understanding his theorem that every group is isomorphic to a group of permutations. I understand what isomorphism means but I'm not very clear with the idea of a group of ...
1
vote
3answers
58 views

Is $U = \lbrace (1),(2),(1,2),(2,1) \rbrace$ a permutation? or a permutation of a power set?

Question If for a set $S= \lbrace 1 , 2 \rbrace$ the set $T = \lbrace (1,2),(2,1) \rbrace$ is refered to as a permutation, then how would an alternative set $U = \lbrace (1),(2),(1,2),(2,1) \rbrace$ ...
1
vote
2answers
66 views

Is there a neat way to count this?

What is the number of strings of length $n$ formed by the digits ${1,2,3,4,5}$ such that two consecutive entries are not same and $1$ and $2$ doesn't come consecutively ? I tried inclusion exclusion ...
2
votes
3answers
37 views

Different seating arrangements on a 6 seater [closed]

Six actors sit in a row to have their photographs taken. Romeo and Juliet insist on sitting next to each other. Caesar refuses to sit next to Brutus. Falsta and Puck don't mind where they sit. How ...
5
votes
1answer
100 views

Can someone explain how the Schreier-Sims Algorithms works on a permutation group with a simple example?

Can someone explain how the Schreier-Sims Algorithms works on a permutation group with a simple example? All the books I read have a dense notation that hard to comprehend but a simple and concrete ...
0
votes
1answer
33 views

Why nothing taken from n things can have 1 permutation?

Without repeating if I take 0 thing out of n, how the permutation will be 1 (nP0 = 1)? And what will be that permutation? Even allowing repetition, the result is 1 (n^0 = 1)? How is the permutation ...
0
votes
1answer
23 views

Determinant of the unit matrix proof

I need to prove that the determinant of any unit matrix is 1, using the defition of the sign of permutation. As far as I know the sign is defined as +1 if the permutation is even, and -1 if the ...
1
vote
0answers
58 views

Group Action highly transitive

$\Omega$ is infinite set, $X$ is a primitive permutation group on $\Omega$, then why if $Alt(\Omega) \leq X$ (that is, $Alt(\Omega)$ is a subgroup of $X$), then $X$ is highly transitive?
0
votes
1answer
22 views

Find a permutation and its signature

Can you help me out with this? $ \alpha = (125)(1378)(12546)(126) $ I need to find $\alpha^{77}$ and the signature of $ \alpha $. The solution I came up with is to calculate $\alpha^{2},\alpha^{3}$ ...
2
votes
2answers
50 views

Finding number of elements common in two set of permutation?

How to find the number of common terms in two sets of permutation? I need to find the number of 4-letter words that can be formed out of the letters {a,b,c,d} with the conditions, Letter 'b' ...
3
votes
1answer
101 views

Permutations of $10$ people with $3$ people together

In how many ways can $10$ people stand in a row such that $3$ of them are always together? I got it as $7!$ ways because three people are together. Then those three can be put in $3!$ ways. So, the ...
1
vote
2answers
32 views

Abstract Algebra Elementary Properties of Groups

This is Excercise 4.A.5 from Pinter's "A Book of Abstract Algebra": Let $a$, and $x$ be elements of a group $G$. Solve for $x$ in terms of $a$. Solve Simultaneously: $x^2 = a^2$ and $x^5 = e$ ...
0
votes
1answer
70 views

There are 11 members in a family out of which there are 4 males and remaining females

There are 11 members in a family out of which there are 4 males and remaining females. The family has hired three cars for a trip to zoo. The members are to be seated in the cars in such a way ...
1
vote
2answers
72 views

In how many ways can 4 red balls and 7 blue balls be arranged in 3 boxes

In how many ways can 4 red balls and 7 blue balls be arranged in 3 boxes where each box must contain at least 1 red ball and each box can contain less than or equal or 4 balls, under the ...
1
vote
1answer
23 views

If $A_6$ acts on set of size $45$, then every involution fixes five points

Let $G \cong A_6$ and suppose that $G$ acts as a transitive permutation group on a set $\Omega$ of size $45$. I want to prove that every involution fixes five points. Any ideas how this could be done? ...
2
votes
2answers
70 views

How many possible combinations in $7$ character password?

The password must be $7$ characters long and it can include the combination: $10$ digits $(0-9)$ and uppercase letters $(26)$. My Solution: Thus in total there are $7$ slots, each slot could be ...
0
votes
2answers
28 views

Clarification of “distinct” for Combinatorics and counting

This may be a trivial question but I am quite stuck on interpreting combinatorics vs permutation in some cases. For example, suppose the following question: "Seven different gifts are to be ...
0
votes
0answers
22 views

Composition of cyclic permutation

I have a permutation $ \alpha = (125)(1378)(12546)(126) $. I know that I have to follow this rule when calculating permutation in their cyclic form: Ex for the first term $$ 1 \rightarrow 2 ...
2
votes
1answer
63 views

Maximum number of pieces of pizza when making 7 cuts

If we have a circular pizza then the maximum number of pieces we can get by making $7$ cuts in it? The fact that I know the solution only got me the way to find it but it was like a kid trying to ...
1
vote
0answers
25 views

If $PSL_2(2^k)$ acts on odd set such that $|\mbox{fix}(g)|\le 3$ for $g\ne 1$, then $G_{\alpha}$ has element of order $\frac{q-1}{\gcd(q-1,3)}$

Let $G \cong PSL_2(q)$ where $q = 2^k \ge 8$. Suppose $G$ acts as a transitive permutation group on $\Omega$ such that $|\mbox{fix}(g)| \le 3$ for nontrivial $g \in G$. Assume $|\Omega| = |G : ...
3
votes
1answer
38 views

Embedding finite groups in symmetric groups whilst preserving conjugacy

Let $G$ be a finite group and suppose that $f,g \in G$ are not conjugate in $G$. It is classical result that every finite group embeds into a finite symmetric group $S = S_{|G|}$. My question is: can ...
-1
votes
1answer
162 views

How many rectangles can be observed in the grid? [duplicate]

In a $2 × 4$ rectangle grid shown below, each cell is a rectangle. How many rectangles can be observed in the grid? My attempt : I found a formula somewhere, Number of rectangles are $= ...
0
votes
0answers
27 views

In permutation group with $3$-group $H := G_{\alpha}$ and $|\mbox{fix}(H)| = 3$ and regular normal subgroup $N$, we have that $C_N(H)$ has even order

Let $G$ be a finite transitive permutation group on $\Omega$. Suppose that $G_{\alpha}$ is a $3$-group and that $|\mbox{fix}_{\Omega}(G_{\alpha})| = 3$ and that $G$ contains a regular normal subgroup, ...
1
vote
3answers
76 views

How many ways to put 20 things to different 4 boxes?

I have 20 identical balls. I want to put these 20 balls to 4 different boxes. In how many ways I can do it? (If necessary we can keep one or more boxes empty)
0
votes
2answers
19 views

Equiprobable model combinations

In a question in our statistics project, there is a set of balls, numbered $1$ to $10$, each ball is equally likely to be selected, making the sample space $S = \{\{i, j, k, l\} : 1 \leq i, j, k, l ...
2
votes
1answer
41 views

Combinatorics/Probability Insurance accident

An insurance company classifies people as normal or accident prone. Suppose that the probability that a normal person has an accident in a specified year is 0.2 and that for an accident prone person ...
0
votes
2answers
19 views

Sign of a permutation including a trivial cycle

This may be a rather basic question, but I can't see mention of this anywhere. Suppose I have a permutation $p\in S_5$ (say). Suppose further that $p$ decomposes as $p=(1 2)(3)(4 5)$. What is the ...
0
votes
0answers
23 views

If $e \in E$ where $E$ is a component of a permutation group, then $e$ has order $2$ or $3$ if $|\mbox{fix}(g)| \le 3$ for all nontrivial $g \in G$

Suppose $G$ is a transitive permutation group such that all four-point stabilizers are trivial, and $G$ has some nontrivial three-point stabilizer. Said differently this means that every nontrivial ...
0
votes
1answer
26 views

Curve connecting two points in $\mathbb{R}^n$ passing through a hyperplane

Let $\pi$ and $\lambda$ be two distinct permutations of $1, 2, . . . , n$, and consider the points $p := (\pi(1),\pi(2), ... , \pi(n))$ and $r:= (\lambda(1), \lambda(2), ... , \lambda(n))$ in ...
-3
votes
1answer
79 views

Expected Value of a in a randomly chosen Rectangle

There is a N×M grid. Each square in the grid either has or does not have a mango tree. For example, suppose the field looks as follows. We Know That there are K Mango Tree. ...
0
votes
0answers
20 views

Finding maximal product of numbers of permutations

Let $n\geq 1$ be a total number of objects that must be taken from $m\geq 1$ sets of objects. For all $i \in \{1,\cdots,m\}, \ M_i \in \mathbb{N}^*$ denotes the number of objects present in the set ...
1
vote
1answer
38 views

Selection of subsets

This is an supplementary exercise from Miklos Bona: A walk through combinatorics. We want to select as many subsets of $[n]=\{1,2,3,..,n\}$ without selecting two subsets such that neither one of them ...
0
votes
1answer
22 views

How to form 3 groups and allocate them different district?

At an election three districts are to be canvassed by $10, 15$ and $20$ men respectively. If $45$ men volunteer, in how many ways can they be allotted to the different districts? Three groups ...
2
votes
1answer
98 views

Number of solutions of $N_9 + N_8 + N_7 + N_6 + N_5 + N_4 + N_3 + N_2 + N_1 = 82$ in the positive odd integers with $N_i \leq N_{i - 1}$

Given $N_{tot}=82$ where $N = [N_9 \: N_8 \:N_7 \:N_6 \:N_5 \:N_4 \:N_3 \:N_2 \:N_1 \:N_0]$, how many possible combinations are there if each $N_i$ must be odd and $N_i \leq N_{i-1}$, i.e. one ...
1
vote
2answers
22 views

Transpositions and the identity permutation

Prove that $\sigma^2$=$\epsilon$ if and only if $\sigma$ can be expressed as a product of disjoint 2-cycles. $\epsilon$ denotes the identity permutation. Any hints to help me get started would be ...