Questions tagged [permutation-cycles]

For elementary questions concerning permutation cycles and permutation groups. This includes all representations of permutations (two-line arrays, cycles, bipartite graphs); transpositions and the sign/parity of a permutation; the Symmetric group and the Alternating group. To be used with the (permutations) tag to make the distinction between abstract algebra permutation questions and combinatoric permutation questions.

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17 views

Let $H$ be a subgroup of $S_n$. Show that if $H$ is a transitive subgroup of $S_n$ and $H$ is generated by some set of transpositions, then $H = S_n$ [duplicate]

So I am convinced that I am misunderstanding the question and it's probably due to the fact that I am misunderstanding the definition of $H$ being a transitive subgroup of $S_n$ Since $H$ is a ...
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16 views

Prove that a cycle of length $k\geq 2$ can be written as a product of $k-1$ transpositions.

Prove that a cycle of length $k\geq 2$ can be written as a product of $k-1$ transpositions as follows: $$ (a_1 ... a_{k-1} a_{k})=(a_1 a_{k})(a_1 a_{k-1})...(a_1 a_2).$$ I found an answer here: ...
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47 views

What are the cyclic permutations on four symbols?

I am trying to solve a problem of Hamermesh's Group Theory and its Applications to Physical Problems. Problem: The cyclic permutations on four symbols form a subgroup $H$ of $S_4$. Resolve $S_4$ into ...
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Analytic estimates of limit cycle parameters

Suppose we have a two-dimensional system of differential equations, say, the well-known Van der Pol oscillator: $$ \dot{x}=y, \dot{y}=\mu (1-x^2)y-x $$ Everyone knows that the study of limit cycles ...
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38 views

is there any way to find the maximal length of cycle in a permutation group?

How we can find the biggest cycle of a permutation group with given the set of generators ? is there any algorithm or theorem for that? Or how we can find the set of generators with biggest cycle as ...
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How an order of permutation can be defined?

Let $\sigma$ be the permutation: [ \begin{array}{lllllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ 3 & 5 & 6 & 2 & 4 & 9 & 8 & 7 & 1 \end{...
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Cyclic order and cyclic permutation

The definitions of a cyclic permutation I know do not use the notion of a cyclic order: Definition of cyclic permutation https://en.wikipedia.org/wiki/Cyclic_permutation https://mathworld.wolfram.com/...
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How does the cycle structure of an invertible linear map over $\Bbb Z/n\Bbb Z$ compare to its affine translations?

The general question I am interested is roughly what is in the title; more concretely: Fix a positive integer $n$, and $m\in(\Bbb Z/n\Bbb Z)^\times$. For each $b\in\Bbb Z/n\Bbb Z$, define the ...
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What, if anything, do permutations have to do with order?

My course notes have: $$ \text{"A bijective function } f:X\rightarrow X \text{ is called a } permutation \text{ of } X" \tag{0}. $$ So let $$A=\{1,2,3\} \tag{1.1},$$ $$f:A\rightarrow A \text{ be a ...
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How to prove that square of a permutation in a cyclic form is even [duplicate]

I want the proof for the above result I tried to prove by using proof by cases where the cycle is an n-cycle and what if n is even or odd
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An Introduction to the Theory of Groups 1.13

The question asks: if $\alpha$ is an $n$-cycle, then $\alpha^k$ is a product of $gcd(n, k)$ disjoint cycles, each of length $n/gcd(n, k)$. I am not quite sure how to proceed with the problem, because ...
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decide $\sigma^{2020}$ [closed]

I don't know how to do this. Please help me.
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66 views

Solution verification: How many elements of order 3 are there in $S_5$

How many elements of order 3 are there in $S_5$. I found it a several ways to solve this one. I did like this, but do not know is it correct: I have a possible options (a b c) = ( b c a) = ( c a b)...
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47 views

Let $G=\langle a,b\rangle$. Prove neither $\langle a\rangle$ nor $\langle b \rangle$ are normal in $G,$ for $a=(1234)(57), b=(24)(5678)$.

Let $G=\langle a,b\rangle,$ a group formed by two permutations of $S_8$: $$a=(1 2 3 4)(5 7)\quad\text{and}\quad b=(2 4)(5 6 7 8).$$ I have to prove that neither $\langle a\rangle$ nor $\langle b \...
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Number of Cycle in Permutations of 2D Grid

Let's consider the following permutation, $7, 3, 1, 2, 6, 5, 4$ Here, we find $2$ cycles: Starting from the first position, position $1$ has the value $7$, position $7$ has the value $4$, ...
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Find a cylic subgroup of order $6$ inside $S_{5}$ . Find all the abelian subgroups of $S_{5}$ .

I am not sure how to approach this question. I know a subgroup of order six would be $(123)(45)$, but I'm not sure how this would be related to abelian groups. I imagine every transposition is abelian....
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Consider a necklace that can be rotated and turned over. It is a circular necklace with 12 identical beads.

Consider a necklace that can be rotated and turned over. It is a circular necklace with 12 identical beads. Let the symmetry group be denoted by G. (i) Determine the number of rotations G contains? ...
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Showing that $f(x) = 4x^2 - 3x^7$ is a pemutation in $\mathbb{Z}_{11}$ and factoring it as a product of disjoint cycles

I need to show that $f:\mathbb{Z}_{11}\to \mathbb{Z}_{11}$ given by $f(x) = 4x^2 - 3x^7$, where the operations are the usual in $\mathbb{Z}_{11}$, is a permutation of the elements of $\mathbb{Z}_{11}$....
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Sum of Traces of all Permutations of a Matrix

I have a problem that reduces to finding the sum of the traces of all possible row permutations of a large matrix. For example if we had a 3x3 matrix, the possible row orderings would be (1,2,3),(1,3,...
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Find element of Group

Let$ (1 3 5 7 )$ and $(2 3 6 8)$ be elements of $S_8$. Find a element$π$ form $S_8$ for which it is worth $π (1 3 5 7 ) π^{-1} = (2 3 6 8)$
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Prove for $\sigma \in S_n$, $\sigma(a_1, … , a_n)\sigma^{-1}= (\sigma(a_1),…, \sigma(a_n))$ [duplicate]

How do I even begin this problem? Does it involve induction or something like induction? Would it be more helpful to instead prove it as $$\sigma(a_1,..., a_n) = (\sigma(a_1),...,\sigma(a_m))\sigma\ ?...
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All permutation groups acting on a finite set are closed

I'm reading through Dixon and Mortimer. Exercise 2.4.3 asks you to prove that if $|\Omega|<\infty$, then every subgroup of $\hbox{Sym}(\Omega)$ is closed. The definition of a closed permutation ...
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Determine all quotient group of $S_3$ and the corresponding homomorphism image.

The group $S_3$ has only three normal subgroups. They are $S_3, \{e\}$, and $\lbrace e,(1 \ 2 \ 3), (1\ 3 \ 2) \rbrace$ where $e$ is the identity element of $S_3$. Determine all quotient groups of $...
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52 views

The number of inversions of an arithmetic progression modulo some number

Let $n, m$ be coprime and consider the arithmetic progression $0,n,2n,\dots$ modulo $m$. As $n,m$ are coprime, the sequence repeats itself with a period of $m$. We can also see this sequence as a ...
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$\beta^k $ is a cycle ⟺ gcd($k$ , o($\beta$))=$1$

GIVEN: $\beta$ is a cycle and belongs to $S_{n}$, then $($$\beta$$)^k $ is a cycle iff $(k,o($$\beta$$))=1 $ let o($\beta$)=m and $ $$\beta$$=(a_{1} a_{2} .... a_{m}) $ While trying to prove the ...
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197 views

How many cycles in this adjacency matrix?

Consider the $N\times N$ adjacency matrix $A$ such that $$A_{ij}=\begin{cases} 1, &\; \; (i\leq n \;\text{ or }\; j \leq n), \;\; i\neq j \\ 0, & \; \; \text{otherwise}\end{cases} $$ for some ...
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58 views

Prove that $Z(S_n)=\{1\}$ for all $n\geq 3$.

I'm going to type down here the answer I found in a manual solutions. However I have a question about it. And I hope I can get help here on this site. Well, I saw that there are several other issues ...
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Prove that the inverse permutation of transposition is itself a transposition. [closed]

Prove that the inverse permutation of transposition is itself a transposition.
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Calculating a product of 3-cycles

I am trying to calculate the following cycle permutation: $$(x_d,x_a,x_c)^{-1}(x_c,x_e,x_b)^{-1}(x_d,x_a,x_c)(x_c,x_e,x_b).$$ But I yield the wrong answer; according to the proof (the complete proof ...
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31 views

There exists a group element $\sigma\in S_7$ under some conditions?

It is given that $\sigma\in S_7$ where $S_7$ is a symmetric group. Do there exist an element $\sigma$ such that $\sigma^{20}=\sigma$ and $\sigma\ne e$ where $e$ is an identity element? My attempt: ...
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How to prove there is no order $15$ permutation in $S_7$?

How to prove there is no order $15$ permutation in $S_7$? I know that there is no element of $S_7$ that has a product of disjoint cycle with order $3$ and order $5$. However, I don't know how to ...
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25 views

How to find the number of left coset of $H$ in symmetric group $S_7$

It is given that $\sigma=\,(124)(3567)$, let $H$ be the subgroup of symmetric group $S_7$ generated by $\sigma$. Find the number of left coset of $H$ in symmetric group $S_7$. My attempt: I tried ...
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Choice of generator in dihedral group

I'm about to begin studying group representation theory, and I want to get more familiar with the symmetric group $\mathfrak{S}_n$ (and its subgroups) first. In particular, I'd like to better ...
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In the Burnside book “Theory of Groups of Finite Order” — error on page 3?

I just got "Theory of Groups of Finite Order" by Wm S. Burnside (1897,1911). Googling [ burnside "ac followed by ab" ] gives many links to page 3, Chapter I "On Permutations". Item 6 says "the ...
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Let be $\sigma=(1\cdots n) \in S_n$. Then $C_{S_{n}}(\sigma) = \langle \sigma \rangle$. [duplicate]

Beggining of the proof: Note that every power of $\sigma$ comutes with $\sigma$, then $\langle \sigma \rangle \subset C_{S_{n}}(\sigma)$. I have two questions: Why $|\langle \sigma \rangle|= o(\...
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How to proof (step-by-step) that [closed]

If $g \in S_n$ and let be $1\leq k \leq n$. Prove that $g(12\cdots k)g^{-1}=(g(1)g(2)\cdots g(k))$. In $S_n$ the cycle $(12\cdots k)$ means $1\longmapsto 2$ $2\longmapsto 3$ $\vdots$ $k-1\...
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Distribution of balls into boxes.

I came upon this questions on one of my tests, and wasn't able to solve it. Was wondering if anyone could point me in the right direction. Thanks in advance. Let m be the number of ways 3 identical ...
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20 views

Prove that every element of $S(\mathbb{Z})$ can be written as a product of at most two elements of order 2 [duplicate]

Prove that every element of $S(\mathbb{Z})$ can be written as a product of at most two elements of order 2 The hint given: First show that a cyclic permutation is the product of two elements of order ...
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Simple permutation multiplication of two 4-cycles from $S_4$, specifically, $ (1324)(1423)$.

For some reason I am confused on how to multiply these together with $4$ elements. I can do it with $2$ elements but theres a little gap in my logic and I am sure one of you can help me with a simple ...
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How does one prove that the decrement of a permutation is equivalent to the signature of the permutation

The decrement $d$ is defined as $n-c = d$, where $n$ is the number of elements in the group and $c$ is the number of independet cycles of certain permutation. If $d$ is even (odd), then the claim is ...
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57 views

The Number of permutations of [n] such that $|\pi(i)-i|\leq 2$

Let $a_n$ be a number of permutations of the set $[n]$ such that $$ |\pi(i)-i|\leq 2$$. I can obtaine the GFs of these restriction permutations. It is equal $$F(z)=\frac{1-z}{1-2z-3z^3+z^5}$$ I ...
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54 views

How many ways to paint $n$ windows?

Given a building with $n$ floors and a window on each floor. You need to paint all the windows either black, white or red. How many ways are there to paint all the windows given that there have to be ...
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Showing that $H$ is a normal subroup of $A_4$

Consider the subgroup $H$={$(1),(12)(34),(13)(24),(14)(23)$} How would I show that $H$ is a normal subgroup of $A_4$? If I perform a few computations such as: $$(132)^{-1}(12)(34)(132)=(123)(12)(34)...
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23 views

Size of the conjugacy class of the element σ = (1, 2, 3)(4, 5) in the group Sym5 . Order of the centraliser CSym5 (σ)

I am a little confused with this question. Am i right in looking for all the elements in Sym5, that are conjugate to $\sigma$ , in other words $a = b\sigma b^{-1}$ for $a,b \in $ S5 ther will be 5 ...
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31 views

Help with permutation (13)(246)(1524)

I am trying to compute a permutation: $\alpha = (13)(246)$ and $\beta = (1524)$. Find $\alpha\beta$. I thought that I had a good grasp on these, but I checked it on a site, and if it is correct, I ...
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445 views

How to know if a 8 puzzle is solvable

Consider the following $3\times3$ sliding puzzle: This is not solvable from the following state: The explanation is that there are 11 inversions, and therefore it is unsolvable. How do they arrive ...
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1answer
56 views

Differences of Even Permutations and Latin Squares

a.) Let $n$ be even. Find a permutation $x_1, x_2,\dots, x_n$ of the elements of $\mathbb{Z}_n$ such that the differences $x_{i+1} - x_i, 1 \leq i < n$, are all different. Say if I take $n=6$, ...
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36 views

Sufficient condition that a permutation is even

Let $a$ be a permutation of $S_n$ and suppose that $a$ decomposes as a product $a = \sigma_1 \circ \dots \circ \sigma_r \circ \gamma_1 \circ \dots \circ \gamma_s$ of disjoint cycles where $\sigma_i$...
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91 views

When $\langle \sigma\rangle$ and $\langle\tau\rangle$ intersect trivially, where both $\sigma$ and $\tau$ are $n$-cycles in $S_n$

Let $\sigma,\tau\in S_n$ be two $n$-cycles. When does $\langle\sigma\rangle\cap\langle\tau\rangle=1$? Note that $\sigma$ and $\tau$ are conjugate in $S_n$ and WLOG we may assume $\sigma = (1,2,\dots,n)...
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1answer
26 views

Prove the map has a 10-cycle but no 6-cycles

Given the map $\tilde{f}:[1,13]\to[1,13]$, and $\tilde{f}(x)=\begin{cases} f(x)+8;& 1\le x\le 5\\ x-8;& 9\le x \le 13\\ \end{cases}$, which looks like the figure: Prove that the map has a 10-...

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