9
votes
3answers
117 views

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} ...
0
votes
1answer
29 views

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 ...
1
vote
0answers
32 views

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 ...
10
votes
2answers
157 views

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 ...
4
votes
2answers
69 views

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 ...
4
votes
1answer
68 views

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 ...
1
vote
3answers
68 views

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 $S_5$.
0
votes
0answers
34 views

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 ...
4
votes
2answers
117 views

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 ...
3
votes
3answers
168 views

Do row and column permutations generate all permutations?

Suppose $m,n\ge 1$ are integers. Do row and column permutations of an $m\times n$ matrix generate the group of all permutations of the $mn$ entries of the matrix? More formally, let $A_1$ be the ...
0
votes
0answers
23 views

Comparing test statistics and/or $p$-values from multiple permutation (Mantel) tests on non-independent data

I am comparing the relationship between genetic and geographic distance of individuals in a wild animal population. The hypothesis is that individuals with higher genetic relatedness establish home ...
4
votes
1answer
32 views

Any two $n$-cycles are conjugate in $A_{n+2}$ if $n$ is odd

How would one go about proving the claim in the title? I see that if $\alpha,\beta$ are $n$-cycles and $\alpha,\beta$ permute $A,B\subset \{1,\dots, n+2\}$ respectively then $\overline{A\bigcap B}$ ...
0
votes
2answers
31 views

Finding a cycle with a specific property

I am reading the book Dummit and Foote - Abstract Algebra . One of the exercises is to find an $n$-cycle $(n \ge 5)$, $\sigma$ such that $\sigma^k = \tau$ for some positive integer $k$, where ...
1
vote
1answer
19 views

Permutation cycles in Jacobson's Basic Algebra I.

Nathan Jacobson's Basic Algebra I Second Edition, Section 1.6 Cycle Decompositions of Permutations, page 51, exercise 4 says: Show that if $\alpha$ is any permutation then $$\alpha (i_1 i_2 \cdots ...
0
votes
2answers
18 views

Find out in how many ways the operation can be performed?

i) In how many ways can a committee of $5$ or more be formed from 12 persons? ii) In how many ways can a committee of $5$ be formed from 12 persons if only two of a group of $3$ persons must always ...
0
votes
2answers
36 views

$(34)(123)(456)$ is a cycle. True or False?

I know this is basics, and I understand that $(34)(123)(456)$ is a product of cycles which, I found: $(124563)$. But somehow, I was lost. How do I know if it is indeed a cycle? OR if it isn't? Any ...
2
votes
2answers
73 views

The Cayley Representation Theorem.

This theorem states that "Any group is isomorphic to a subgroup of a group permutations." I only ask if someone could provide a simple example so that i can fully understand this theorem.
0
votes
1answer
42 views

Conjugate subgroups of $S_4$

$A = \langle (1,2,3),(1,2)\rangle$ $B = \langle (1,2,4),(1,2)\rangle$ $C = \langle (1,3,4),(1,3)\rangle$ $D = \langle (2,3,4),(2,3)\rangle$ I want to proof that these subgroups of $S_4$ ( which ...
1
vote
1answer
44 views

How many distinct elements does a group of permutation on 3 letters have?

I am having some problems solving a problem similar to this. So i tried making it a more simpler problem. I really don't know how to approach this kind of problem. A hint would be very much ...
1
vote
3answers
49 views

Writing a permutation group in 2 row notation

I have a permutation group in $S_7$, namely: $$(12345)(137)(56)$$ How do I write this in two row notation? I am to write it as disjoint cycles and then as transpositions but I feel better working in ...
2
votes
2answers
40 views

Kernel of $\phi:G \rightarrow \operatorname{Sym}(S)$ Group actions

$\operatorname{Sym}(S) == \text{All permutations of the set }S$. Prove $\ker(\phi)=\bigcap_{x\in S}G_x$ where $G_x$ is the stabilizer of $x$. Let $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
2
votes
0answers
29 views

How many commutative block ciphers are there?

Let $K$ and $M$ and be two finite sets. Let $(G,\circ)$ be the group of permutations over $M$ under composition. Let a (implicitly: block) cipher with key in $K$ and message in $M$ be any application ...
14
votes
1answer
169 views

Involutions, RSK and Young Tableaux

Let $S_n$ be the symmetric group on $n$ elements. The Robinson-Schensted-Knuth (RSK) correspondence sends a permutation $\pi\in S_n$ to a pair of Standard Young Tableaux $(P,Q)$ with equal shapes ...
0
votes
1answer
17 views

Prove that cyclic index of this operation can be expressed by formula

Let $T_1$ and $T_2$ be disjoint finite sets and let $G_1$ and $G_2$ be, respectively, some groups of permutations of this sets. Direct sum $G_1 \bigotimes G_2$ acts on $T_1 \cup T_2$: $$ \langle ...
2
votes
1answer
53 views

Finding the smallest positive integer $n$ such that $S_n$ contains an element of order 60.

I am trying to find the smallest positive integer $n$ such that $S_n$ contains an element of order 60. I know that every permutation in $S_n$ can be expressed as the product of disjoint cycles, and I ...
0
votes
1answer
47 views

Showing that there is a permutation $\rho$ that fixes a number that $\sigma$ moves when $\rho \sigma \rho^{-1}=\sigma^{-1}$

Doing an assignment, getting a bit frustrated with this exercise, would really appreciate some help. The first exercise explains what $\sigma$ and $\rho$ are: Let $\sigma$ be the $r$-cycle ...
0
votes
1answer
149 views

How many natural numbers not more than 4300 can be formed with the digits 0,1,2,3,4 (if repetitions are not allowed)?

My Approach: Total no of available digits: 5 No of 1-digit formed: 5 No of 2-digit formed: 4(excluding '0' on ten's place) * 4 (including zero - one among given digits already used) No of 3-digit ...
1
vote
1answer
20 views

What are the number of circular arrangements possible?

Suppose we have $4$ identical red beads and $3$ identical blue beads. In how many ways can we form a necklace out of these? I am a little confused here. Suppose we fix a red bead and treat the ...
2
votes
1answer
47 views

Signature of permutations is a homomorphism

Given the following definition of $signature$: $\epsilon(\sigma)=(-1)^{n-k}$, where $k$ is the number of cycles (with disjoint supports, counting the 1-cycles) of the permutation, prove that ...
0
votes
1answer
58 views

Odd Permutations

Prove that the product of two odd permutations is even. I'm having a difficult time doing this in the general case. I have that if s is even, then $$\alpha = ...
2
votes
1answer
28 views

Permutation query

Would anyone be able to help here with this one ? Let $A = \{a, …, z, A, …, Z, 0, …, 9\}$ be some alphabet and let $$q = q_1, …, q_m \text{ and } w = w_1, …, w_n$$ be finite-length words in $A^*$. ...
0
votes
1answer
64 views

How many permutations of the sequence 1, 2, 3…N where none of the first K numbers in the original sequence is in it's place?

For the sequence 1, 2, 3 ... N there are of-course N! permutations. But for a given K, where 1 < K ≤ N how many permutations are there given none of ...
0
votes
0answers
36 views

A conjugation of an operator, which commutes with all permutations, still commutes with all permutations

Assume $v:H^{\otimes m}\to H^{\otimes m}$ is a linear operator on the $n^{\text{th}}$ tensor power of a vector space $H$. For each permutation $p$ on the $m$-element set define the linear operator ...
1
vote
3answers
78 views

Number combination with groups and the smallest repetition possible

I need to create a equation to assign a number of phrases (variable A) to a a defined number of groups (variable B) and repeat these assignment each day, and repeat this operation along of time with ...
1
vote
2answers
70 views

Permutation Group-$S_{10}$

How many elements of order $30$ are there in the symmetric group $S_{10}$? I worked out and got $10500$ - using Computations and adjusting each individual cycle's position ( $2-3-5$ , $3-2-5$ etc). ...
4
votes
0answers
94 views

Wikipedia's Cayley Table and Pictures for 3 by 3 Permutation Matrices

Are there any explanations or clarifications of the pictures at https://en.wikipedia.org/wiki/Permutation_matrix#Permutation_of_rows_and_columns? ...
4
votes
2answers
128 views

How many elements of order $k$ are in $S_n$?

I need to find how many elements of order $k$ are in $S_n$ (where $k \leq n$). So if $k$ is prime, it's easy: $k$ can't be the $\mathrm{lcm}$ of any integers besides itself and one's (which we're ...
3
votes
1answer
91 views

Existence of a solution for an equation in a permutation group

Here is a concrete example, but I'm looking for methods in general : Let $S_{13}$ be the permutation group. Let $i : S_2 \times S_3 \times S_4 \times S_4 \to S_{13}$ be the canonical injection. Let ...
2
votes
2answers
159 views

Secret Santa Combinatorics with couples

I have researched this site and find several secret Santa related questions but none that I can find that relates to couples. If there are three couples (6 people) and no one can draw their own name ...
-1
votes
1answer
119 views

Arranging n different object in k groups [closed]

how many ways we can arrange $n$ different objects in $k$ group. Like for $k$=1 our answer will be $n!$
2
votes
1answer
65 views

How to show that an automorphism of $S_n$ is inner?

Given an automorphism $\phi:S_n\rightarrow S_n$ such that it maps all the transpositions on the transpositions, how do I show that this map is given by a conjugation with an element $s\in S_n$? ...
1
vote
1answer
31 views

Product of non-disjoint k-cycles

I have two $k$-cycles $\alpha=(a \dots c \dots b \dots)$ and $\beta=(a \dots b \dots c \dots)$ and $\alpha \neq \beta^{-1}$. How to show that the product $\alpha \beta$ does not result in a cyclic ...
0
votes
1answer
77 views

Properties of Permutations of a Set A

Let $A$ be a finite set, and $B$ a subset of $A$. Let $G$ be the subset of $S_A$ consisting of all the permutations $f$ of $A$ such that $f(x)\in B$ for every $x\in B$. Prove that $G$ is a subgroup of ...
1
vote
0answers
126 views

Assumption in proof: The alternating group $A_n$ is simple.

Let $N$ be a nontrivial normal subgroup of $A_n$, $n \ge 5$, where $A_n$ denote the alternating group. The book want to prove that $N$ contains a 3-cycle. (Niels Lauritzen, Concrete Abstract ...
2
votes
1answer
110 views

Does a homomorphic image of even permutations consist of even permutations?

If $f:S_n \to S_n$ is a homomorphism, prove $f(A_n) \subseteq A_n$. If every image of a transposition is even, then there is nothing to prove, but it is not sure.. How can I prove the problem?
1
vote
1answer
74 views

Order of a product of two cycles

Is it true or false? If $a$ is a permutation that is an $m$-cycle and $b$ is the permutation that is $n$-cylce then order of $ab = \operatorname{lcm}(m,n)$.
2
votes
0answers
82 views

Number of permutations possible?

Given two permutation of $1, \ldots, N$. Where 3<=N<=1000 Example For $N=4$ First is $\begin{pmatrix}3& 1& 2& 4\end{pmatrix}$. Second is $\begin{pmatrix}2& 4& 1& ...
0
votes
1answer
102 views

Every nontrivial subgroup $H$ of $S_9$ containing some odd permutation contains a transposition. [duplicate]

This is a true or false question. Apparently, it is false, but I don't follow. Clearly, if it contains an odd permutation, and an even/odd permutation is defined by the number of transpositions it ...
2
votes
2answers
110 views

Size of alternating group $A_n$

This is not too obvious to me - what is the size of alternating group? Following the hint in the comment, should it be $A_n = S_n/2$? So I don't feel right up to here.....
1
vote
1answer
196 views

Theorem regarding greatest common divisor of certain Binomial coefficients.

Recently my friend asked following question- find the greatest common divisor of all binomial coefficient for a given n so the problem is in mathematical form ...