# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 $S_5$, ...
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### 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 $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S$$ ...
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### 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 ...
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### 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, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, $\langle (12\cdots n),(12)(34)\rangle =\mathcal A_n$. ...
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### 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 ...
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### 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 ...
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### 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. ...
### 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) ...