1
vote
1answer
48 views

number of cycles in a permutation

I have given a permutation let 2 3 1 5 4 that is if initially my string is 1 2 3 4 5 the after one permutation it will become 1 2 3 4 5 3 1 2 5 4 that is the number in first position will go ...
1
vote
0answers
31 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 ...
11
votes
2answers
144 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
63 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 ...
3
votes
1answer
65 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 ...
3
votes
2answers
42 views

Working with groups. Finding the inverse of some $S_9$

I want to compute the inverse of: $\begin{pmatrix} 1&2&3&4&5&6&7&8&9\\3&2&1&6&5&9&4&8&7 \end{pmatrix}$ Sorry about alignment(they are ...
1
vote
3answers
65 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
1answer
20 views

Conjugating a permutation

I am trying to see that two permutations are conjugate exactly when they have the same cycle decomposition. I fail to see that $$r(i_1,i_2,\dots,i_k)r^{−1}=(r(i_1),r(i_2),\dots,r(i_k))$$ Any thoughts ...
1
vote
1answer
14 views

Group of permutations and disjoint cycles.

Two cycle $[i_1,..., i_r]$ and $[j_1,...,j_s]$ are said disjoint if no integer $i_\eta$ is equal to any integer $j_{\mu}$. Prove that a pemutation is equal to a product of disjoint cycles. I was ...
4
votes
2answers
110 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
159 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 ...
3
votes
3answers
49 views

Conjugate to the Permutation

How many elements in $S_{12}$ are conjugate to the permutation $$\sigma=(6,2,4,8)(3,5,1)(10,11,7)(9,12)?$$ How many elements commute with $\sigma$ in $S_{12}$? I believe I use the equation ...
0
votes
1answer
42 views

Why do we swap the position in the cycle when writing disjoint cycles

I was watching a video on tranpositions and it isn't obvious to me why when decomposing a cycle, we swap the position of the elements in the cycle instead of swapping the elements themselves. I would ...
0
votes
1answer
33 views

All permutations from $S_6$ and $S_7$ by which $(1,2)(3,4,5)$ is conjugate to itself

The task is to find all permutation $\tau$ from $S_6$ and $S_7$ such that: $$\tau^{-1}(12)(345)\tau=(12)(345)$$ I think the answer is: $\{id \in S_6 , id \in S_7 , (67) \in S_7\}$ I would just like ...
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 $\overline{A\bigcap B}$ ...
1
vote
2answers
38 views

How to find the elements of $\langle (1,2,3,4) \rangle$ in $S_4$?

How to find the elements of $\langle (1,2,3,4) \rangle$ in $S_4$? The answer is given as $\{\mbox{id}, (1,2,3,4), (1,3).(2,4), (1,4,3,2)\}$. I understand how we got the first $2$ elements. Also ...
1
vote
3answers
68 views

Group theory, conjugation of permutations

I have a past exam question that says... Decompose the following permutations into a product of disjoint cycles. Are the two permutations conjugate? $$\alpha= \begin{bmatrix} 1 & 2 ...
1
vote
1answer
29 views

Find the unknown permutation

Find a permutation $x$ such that For $$y = (1274)(356)$$ $$x^{-1}yx = (254)(1736)$$
0
votes
1answer
49 views

Group Theory, Permutations.

Let $x := (175)(2436)$ and $y := (1234567)$ Compute the elements $x^{-1}yx$ and $y^{-1}xy$? And if $z := (1326745)$ for which $u$ is there an element $v$ such that $v^{-1}zv=u$. If $u$ exists find ...
4
votes
0answers
192 views

Special Products of Transpositions

[Edit. Significantly expanded to add examples and (I hope) clarification. Feel free to skim by reading the gray boxes.] A colleague asked me for insights on a collection of special permutations, ...
0
votes
1answer
27 views

Is commutation transitive for permutations?

Let $f,g,h$ be bijections from some set $X$ to itself, i.e. they are permutations of $X$. We say that $f$ and $g$ commute if $f\circ g=g\circ f$. Is it the case, in general, that if $f$ commutes with ...
1
vote
2answers
57 views

Does the alternating group $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$?

What are all the possible orders of elements in the group $A_5$? Does $A_5$ contain a subgroup isomorphic to $\Bbb Z_{20}$? How about $\Bbb Z_{10}$? How about $\Bbb Z_5$? Justify your answers. I've ...
2
votes
2answers
64 views

How do I prove that an order of a cycle is its length?

Let $\sigma$ be a cycle with length $n$ where $\sigma \in S_m$. How do i prove that $|\langle \sigma \rangle |$ is $n$?
2
votes
2answers
67 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.
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
46 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 $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
1
vote
1answer
53 views

Why is this the method to getting transpositions from disjoint cycles?

I have the disjoint cycle: $$(156)(2437).$$ Apparently the "method" would get us: $$(1,6)(1,5)(2,7)(2,3)(2,4).$$ Basically you take the first number, and put it as a transposition of the last number ...
0
votes
1answer
16 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 ...
0
votes
1answer
46 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 ...
1
vote
1answer
47 views

Using that $G$ is isometric to a subgroup of $S_G$ to prove something about $G$

I am doing the following exercise for an assignment: Assume that $G$ is any finite group with non-trivial elements such that $bab^{-1}=a^{-1}$. Let $k$ be a natural number and use induction to ...
0
votes
2answers
22 views

every k cyclic is a product of at least k-1 distinct tranpositions

There is a theorem says if $A$ in $S_n$ is a $k$ cycle, and $A = a_1 a_2 a_3 \dots a_m$, where $a_i$ are transpositions, then $m \geq k-1$. But how to show there are at least $k-1$ distinct ...
1
vote
1answer
52 views

What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;] Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn) ...
2
votes
2answers
92 views

Find the center of the symmetry group Sn.

Find the center of the symmetry group $S_n$. Attempt: By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\ g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is ...
0
votes
1answer
70 views

How to show that a permutation form a group?

Given the following $12$ permutations: $\{(1), (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(32)\}$ (a) Show that the 12 permutations form a group. (b) Find ...
2
votes
0answers
18 views

Clarification on a cycle parity proof

Prove that cycle $(a_1a_2...a_k)$ is even $\iff$ $k$ is odd. This makes intuitive sense because $(a_1a_2a_3...a_k)=(a_1a_k)(a_1a_{k-1})...(a_1a_3)(a_1a_2)$ which will be an even number of ...
1
vote
2answers
57 views

what is degree of permutation group?

Is "degree" the same term as "order" of a permutation group?
1
vote
2answers
43 views

Permutation help

Consider the elements of $S_7$. For each $\sigma \in S_7$ there is a smallest positive integer |$\sigma$| such that $\sigma^{|\sigma|}=e$. Find the value of $N$= max{ $|\sigma|$ | $\sigma \in S_n$}. ...
0
votes
1answer
32 views

Finding a permutation from a power of itself

Find a permutation $\sigma \in S_9$ such that $\sigma^2=(13579)(268).$ So I know that $\sigma^{10}=\sigma.$ But I don't know $\sigma^5$..... Is $\sigma^{10}=\sigma^4\sigma^6$? I doubt this is the ...
2
votes
1answer
44 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 = ...
1
vote
1answer
80 views

What does permutation stand for as a power?

I am just reading some books about abstract algebra and I don't understand what a permutation stands for as a power. For example, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
2
votes
0answers
55 views

Symmetry of the pentagon and even permutation

I was doing part (iii). For the first part of that questions, $ D_{10} = \{e, \rho, \rho^2, \rho^2, \rho^4, \sigma\rho, \sigma\rho^2, \sigma\rho^3, \sigma\rho^4 \} $ where $\rho = (1 \ 2 \ 3 \ 4 \ ...
1
vote
0answers
42 views

Finding a permutation $ \alpha $ given $ \alpha^4 $ [duplicate]

I have the following question: Find a permutation $\alpha ∈ S_7 $ such that $\alpha^4 = (2 1 4 3 5 6 7)$. Is $\alpha$ unique? How should I go about this? I've tried a few different trial and error ...
0
votes
1answer
42 views

permutations and transpositions in even and odd cases

Say we had some $\sigma = (1, 2)(2, 3)...(n-1 ,n)$ could someone explain why this formula doesn't hold for odd n? For instance, $n = 2m+1$ $\sigma = (1,2)...(2m-2,2m-1)(2m, 2m+1)$, why does that not ...
2
votes
2answers
36 views

List of all elements of $A_4$ - Jamie Mulholland p. 85

p. 72: $m$-cycle $\iff m - 1$ transpositions. Hence 3-cycle $\iff 2$ transpositions. I condone all the calculations overhead, but I don't understand the proof blueprint. (1.) How do you ...
0
votes
2answers
53 views

Prove this is a subgroup: Subset of $S(A)$ consisting of all the permutations $f(a) = a$

Let $A$ be a set and $a \in A$. Let $G$ be the subset of $S(A)$ consisting of all the permutations $f$ of $A$ such that $f(a)=a$. Prove that $G$ is a subgroup of $S(A)$. I really have no clue how to ...
2
votes
2answers
69 views

Symmetric Group $S_3$

I just wanted to make sure I am thinking about this correctly. I would like to take the product $(123)(231)$. Here, $2-3-1,3-1-2,1-2-3 \Rightarrow (123)(231)=(132)$.
6
votes
1answer
76 views

Why is the parity of a permutation an important concept?

In Pinter's A Book Of Abstract Algebra, the author states that: A number of great theorems of mathematics depend for their proof (at that crucial step when the razor of logic makes its decisive ...
3
votes
2answers
93 views

Example of a simple graph isomorphic to a permutation group.

I'm taking a first course in graph theory this semester and I'm working trough Graph Theory with Applications by J.A. Bonday and U.S.R. Murty. I can't find an answer to question 1.2.12(f): (a) ...