2
votes
0answers
40 views

Drawing a Truncated Octahedron

I'm trying to draw a truncated octahedron in MATLAB. This is also known as a permutahedron so my strategy is to link up all the vertices via adjacent transpositions of permutations in $S_4$. What I ...
0
votes
1answer
80 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
6
votes
1answer
32 views

Equivalent of a sequence in regard to a certain length of a cycle for $\mathfrak{S}_{n}$

Let $n \in \Bbb{N}$ ( for me $0\notin \Bbb{N})$. Find the limit as $n$ tends to $+ \infty$ of the following sequence $$\frac{\alpha_{n}}{n}$$ where $\alpha_{n}$ is the number of permutations of ...
0
votes
1answer
22 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 ...
3
votes
3answers
52 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 ...
3
votes
1answer
71 views

Recurrence for the number of $\sigma \in S_n$ with cycle length at most $r$

I have just learned that the formula is right, but the definition of $c_n^{(r)}$ was wrong. The correct problem is: Prove $$c_{n+1}^{(r)} = \sum_{k=n-r+1}^n n^{\underline{n-k}} c_k^{(r)}$$ where ...
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.
1
vote
1answer
37 views

Finding the maximum possible order for an element in $S_5$

I understand that you have to write out all the disjoint cycles and then take the least common multiple which yields the highest order. But my question is, do I have to write all elements of $S_5$, ...
2
votes
2answers
41 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
1answer
54 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 ...
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, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
0
votes
1answer
46 views

Order of a permutation using its cycle decomposition

If $A=\{1,2,...,n\}$, $\Omega _A$ is the set of all permutations over $A$, $S_n=(\Omega _A, \circ)$, then for any $\sigma \in \Omega _A$, the order $m$ of $\sigma$ (Smallest $m \in \mathbb{N}$ for ...
3
votes
2answers
136 views

Proving that any permutation in $S_n$ can be written as a product of disjoint cycles

I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes. Proposition: any permutation in $S_n$ can be written ...
2
votes
3answers
100 views

maximum order of an element in symmetric group [duplicate]

While doing my homework i find out that the maximum order of an element in $S_3$ is 3 (the element $(123)$) and the maximum order of an element in $S_4$ is 4 (the element $(1234)$) Can i generalize ...
3
votes
1answer
79 views

Alternating group generators

Consider the alternating group $\mathcal A_n$ ($n$ is an odd integer). Do $(12\cdots n)$ and $(12)(34)$ generate $\mathcal A_n$? In other words, $\langle (12\cdots n),(12)(34)\rangle =\mathcal A_n$. ...
3
votes
2answers
73 views

Conjugation of $3$-cycles in $A_5$

How do you show that any two 3-cycles are conjugates in $A_5$. I know we have to take $2$ $3$-cycles say $a$ and $b$ in $A_5$ then we have to show there exists a $c\in A_5$ such that $a = c b ...
2
votes
2answers
92 views

Find the number of permutations in $S_n$ containing fixed elements in one cycle

Find the number of permutations in $S_n$ such as 1 and 2 belong to one cycle.
1
vote
1answer
124 views

Equal parity of inversions and transpositions

Let $\sigma $ be a permutation of the set $\{1,\ldots,n\}$ that is written as the product of $k$ transpositions. Let $\rm{inv}(\sigma)$ be the number of inversions of $\sigma$ - that is the number of ...
0
votes
1answer
65 views

How do I do cycle notation?

Consider the three permutations given in cycle form: $f = (124)(35)$, $g = (13)(45)(2)$, $h = (1)(4253)$ Which composition of the functions $(f, g ,h)$ gives the permutation $(1)(2)(34)(5)$?
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 $ab = \operatorname{lcm}(m,n)$.
0
votes
2answers
51 views

Groups of Pemutations

I am having a difficult time with the following question: Find a four element abelian subgroup of S5 and then write its table. I am lost as to where to start. Do I arbitrarily choose 4 elements of ...
0
votes
1answer
168 views

Permutations and symmetric groups

Suppose that a permutation $f$ is the product of disjoint cycles $f_1,f_2,\dots, f_m$. Show that $o(f)$ is the least common multiple of $o(f_1), o(f_2),\dots, o(f_m)$. Really lost with the question.. ...
1
vote
2answers
100 views

Approximation for the number of involutions?

I am interested any approximation that may be available for the following expression: $$ {\left(2n\right)}!\sum_{k=0}^{n}\frac {1} {2^k \; k! \; \left(2n-2k\right)!} $$ ... which can be expressed ...
0
votes
2answers
37 views

Name of similar elements of symmetric group

For permutations of length $n$ of the symmetric group $S_n$, they can be arranged as follows (for example, $S_4$): ...
2
votes
2answers
263 views

What is the size of the normalizer of a subgroup generated by a $p$-cycle in a symmetric group?

Question: What is the size of the normalizer of a $p$-cycle (prime $p$) in the symmetric group $S_n$ ($n \geq p$)? If $n<2p$, we can actually find the size $N:=|N_{S_n}(\langle (12\cdots p) ...
3
votes
1answer
83 views

Name of an inversion-like function for permutations

Let $\pi\in S_n$ be a permutation. Define $m_k=|\{i: \ i\leq k, \ \pi(i)>k\}|$. For example, if $\pi=(597216384)$, then it's $m$ vector is $m(\pi):=(m_1,m_2,\ldots,m_{n-1})=(1,2,3,3,2,2,1,1)$. Is ...
1
vote
2answers
107 views

Confused on permutation cycles

I am a bit confused on how to interpret permutation cycle notation. I am going by the Wolfram definition. My initial interpretation of $(431)(2)$ was "4 moves to position 3, 3 to position 1, 1 to 4, 2 ...
0
votes
1answer
69 views

The symmetric group $S_4$ and cycles as product of permutations

i have a question, i think it is very easy but i didn't see it: I know that: $$S_4=\langle(1~~2),(2~~3),(3~~4)\rangle$$ Now i want to write $(1~~2~~3)$ and $(1~~2~~3~~4)$ as products of them. ...
4
votes
5answers
150 views

Confused about the group of permutations $S_{n}$

In an exercise, I must prove $S_{n}$ is generated by 2 elements. I'll ignore here the trivial case $n = 1$. Let $I_{n} = \{1, 2, 3, ..., n\}$. I then defined $f : I_{n} \rightarrow I_{n}$ by $f(1) = ...
2
votes
2answers
102 views

Listing subgroups of a group

I made a program to list all the subgroups of any group and I came up with satisfactory result for $\operatorname{Symmetric Group}[3]$ as ...
6
votes
0answers
116 views

Expression of basis vectors of permutation modules in different bases.

Write $[n]:=\{1,\ldots,n\}$. For a partition $\lambda\vdash n$, I will write $[\lambda]$ for the Specht module that corresponds to $\lambda$, i.e. the complex vector space spanned by all standard ...
1
vote
4answers
70 views

The isomorphism from $S_3/\langle (123)\rangle$ to $\mathbb{Z}_2$.

Suppose that $N=\{(123), (132), \operatorname{e}\}$ and $N$ is normal in $S_3$. Show that the quotient group $S_3/N$ is isomorphic to $\mathbb{Z}_2$. What mapping should I use?
8
votes
1answer
494 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 $\lambda_x (g) = xg$. ...
3
votes
4answers
105 views

Is my textbook wrong?

My textbook says (without explaining how it is done): $$\begin{pmatrix} 1\ 2\ 3\ 4\\ 2\ 1\ 4\ 3 \end{pmatrix}\begin{pmatrix} 1\ 2\ 3\ 4\\ 2\ 3\ 4\ 1 \end{pmatrix}=\begin{pmatrix} 1\ ...
0
votes
2answers
108 views

Product of disjoint cycles question.

Consider the following permutations $x$ and $y$ in $S_6$: $x=(1 \, 3 \, 5)(2 \, 4)$ and $y=(2 \, 3 \, 4 \, 5)$ Express $xy$ as a product of disjoint cycles. My attempt: I first got $xy = (3 \, 5 \, ...
1
vote
2answers
1k views

Cayley's Theorem - Discourse on Fraleigh's proof or Alternative Easier proof [duplicate]

I have hard time understanding Fraleigh's proof. Can someone either explain Fraleigh's or provide an alternative, perhaps easier proof? Thanks so much.
0
votes
1answer
142 views

Anti-Symmetric Complex Polynomial

Let $f(x_1,...,x_n)$ be a complex polynomial. Show the following two conditions on $f$ are equivalent: i) for any transpositions $\tau$ we have $\tau.f=-f$ and ii) for any $\sigma \in S_n$ we have ...
2
votes
2answers
545 views

Efficient method to determine the order of a permutation in $S_n$

Instead of trying multiplication again and again until I get $(1)(2)(3)(4)(5)(6)(7),$ is there an efficient, logical method to compute order of $(157)(134)(12)$ of $S_{10}$? Is there some relation to ...
5
votes
2answers
245 views

Generators of Symmetric and Alternating Group

Consider the symmetric and alternating groups $S_n$ and $A_n$ ($n>2$). 1. Does arbitrary $2$-cycle and an arbitrary $n$-cycle in $S_n$ generates $S_n$? 2. If $n$ is odd, does an arbitrary ...
1
vote
2answers
74 views

Given $\tau\in{S_n} $ a cycle of length $m$ that $\tau^{m} = e$

I'm trying to rigorously prove that given a cycle $\tau\in{S_n} $ of length $m$, then $\tau^{m} = e$ where $e$ is identity. The funny thing is that I know why it works and understand it intuitively ...
3
votes
2answers
376 views

Is every permutation group isomorphic to a 'familiar' group?

This may be a very simple question, but I do not know how to approach it. Cayley's theorem states that every group G is isomorphic to a subgroup of the symmetric group acting on G (in other words, ...
4
votes
1answer
243 views

Wreath Products of Symmetric Groups

I am currently in the process of reading an article by D.Bundy The connectivity of commuting graphs. In section 3 (in the Preliminary Results) Bundy gives the following result: $\mathbf{(3.1)}$ Let ...
1
vote
1answer
201 views

Structure of the centralizer of an element in Sym(n)

Do we know the structure of the centralizer of any element in $S_n$? Or the centralizer of a permutation which has $q$ disjoint $r$-cycles where $r\leq n$. If we know, can anyone give proof or ...
10
votes
4answers
206 views

Group Action - Permutation on the Polynomial

I'm trying to check the permutation on the polynomial is a Group Action, but I'm not getting the second axiom. I'm following my lecturer's work --- Examples 2.1 and 2.6 on page 5 on ...
9
votes
2answers
492 views

normal subgroups of infinite symmetric group

I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the ...
1
vote
1answer
238 views

The sgn function and permutations

Let $P=\{(i, j)|1\leq i<j\leq n\}. $For $\sigma\in S_n$, define $\operatorname{sgn}\colon S_{n}\rightarrow \{\pm 1\}$ by $$ \operatorname{sgn}(\sigma)=\prod_{(i, j)\in ...
0
votes
2answers
159 views

Problems about symmetric groups

We just finished a lesson about determinants and the symmetric group with all what comes with it ( permutations, transpositions etc... ), except we didn't do group theory ( we only see it next year ), ...
2
votes
1answer
46 views

A statement about $S_n$: pigeonhole?

Let $n \in \mathbb N$, $n \ge 2$ and let $S_n$ be the symmetric group on $n$ elements. I will call for shortness $I_n := \{1 , \ldots , n\} \subset \mathbb {N}$. Fix $i_0 \in I_n$ and consider the ...
2
votes
2answers
684 views

Irreducibility of the standard representation of $S_n$.

The permutation representation of $S_n$ is $\mathbb C^n$ with elements of $S_n$ permuting the basis vectors $\{e_1, e_2, \ldots, e_n\}$. It has a trivial subrepresentation spanned by the vector $v = ...
0
votes
1answer
64 views

How many homomorphims from $Z_{12}$ to $S_{5}$

I'm trying to calculate how many homomorphisms exists from $Z_{12}$ $\longrightarrow$ $S_{5}$ . Here are the options : $(x x x x x)$ order 5 $(xxxx)$ order 4 $(xx) (xx) $ order 2 ??? $(xxx)(xx) ...