# Tagged Questions

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### mapping $S_4$ a nd checking if it is cyclic [duplicate]

True or false? Every element of $S_4$ is a cycle. Can anyone help me in how to solve this question? I find difficulty in answering the question.
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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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### 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 ...
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### 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 ...
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### 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 ...
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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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### 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 ...
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### Naming elements of a group

Assume one comes across some set and constructs a finite groups out of these elements. One knows what group it is, names the elements from 1 to order of the group and constructs the multiplication ...
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### 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 $\overline{A\bigcap B}$ ...
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### 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 ...
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### 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 ...
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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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### 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 ...
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### Construction of transitive group of degree $n$

Is there any way to construct all transitive groups of degree 6 with the following block system: {1,2} , {3,4}, {5,6} ?
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### How do I calculate permutations where some values are restricted?

I am curious about the formula for determining the number of combinations there are in a given set where some values are restricted to a certain range. For example, if I have a 10 character, ...
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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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### 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). ...
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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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### Compute number of permutations composed only of transpositions for a given set

Given a set of $n$ elements, how can I find the number of all possible permutations composed only by a product of cycles? For example, for the set $\{1,2,3\}$ there are 4 such permutations: $(123)$, ...
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### 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 ...
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### 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.
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### If $\alpha$ is cycle How do I prove that $\alpha^k$ is cycle?

$ord(\alpha)=r$, $k=\frac{r+1}{2}$, $r$ is odd How I show that $\alpha^k$ is cycle and $ord(\alpha^k)=r$?? Thank you! I add new impotent detail...
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### Question about cyclic (Renew Question) [duplicate]

At my Question about permutation - (I add new details, now it shold be more clear)(Question about permutation - (I add new details, now it shold be more clear) I explain it wrong , now I try to ...
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### Finding the “square root” of a permutation

Suppose $r$ is odd, and ${\rm ord}(\alpha)=r$. ($\alpha,\beta$ are cycles.) Now, $\alpha=(a_1\cdots a_n)$. I need to find $\beta$ that will make $$\alpha = \beta^2$$ How can I show it? What I ...
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### 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 ...
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### Finding the permutation that shows two permutations are conjugates method?

Problem: Given $\sigma=(12)(34)$ and $\gamma=(56)(13)$ find $\tau\in S_6$ with $\tau^{-1}\sigma\tau=\gamma$ Attempt: I'm kind of new to this but from what I understanding find $\tau$ that satisfies ...
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### 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 ...
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### 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 $\lambda_x (g) = xg$. ...
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### How do combinations (not permutations) relate to group theory?

First question. I'm just generally curious about combinations in group theory. How do they relate? If I take the set of permutations of $\langle 1,2,3,4 \rangle$, I get the symmetry group S4. How ...
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### Dihedral group and cyclic group theorem.

Let $D_n$ be the dihedral group defined by $D_n=$ {$I,R,R^2,...,R^{(n−1)},r,rR,rR^2,...rR^{(n−1)}$} Theorem. A nontrivial proper subgroup $N$ of $D_n$ is normal in $D_n$ if and only if $N$ is a ...
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### Subgroups of $A_5$ have order at most $12$?

How does one prove that any proper subgroup of $A_5$ has order at most $12$? I have seen that there are $24$ $5$-cycles and $20$ $3$-cycles. What do the other members of $A_5$ look like?
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### Orders of centralizers $C_G(g)$ in a group of order 60?

Given a group $G$ of order 60 with 24 elements of order 5, 20 of order 3, and 15 of order 2, how do we find the sizes of centralisers of elements of $G$ without proving $G\simeq A_5$? By considering ...
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### Solving conjugacy equations in dihedral groups.

For all integers $m$ such that $0≤m<n$ find $a,b,c\in D_n$ such that $a(rR^m ) a^{-1}=R^2$ $b(rR^m ) b^{-1}=r$ $c(rR^m ) c^{-1}=rR$ $D_n$ is dihedral group of an $n$-gon represented by ...
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### Why is $S_5$ generated by any combination of a transposition and a 5-cycle?

Why is $S_5$ generated by any combination of a transposition and a 5-cycle? Is this true for any prime $p$ (in this case $p=5$)?
How to list the elements of subgroup $\langle a \rangle$ in $S^7$ where $$a=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 7 & 6 & 5 & 1 &4 ... 1answer 87 views ### Conjugation on subgroups of A_4 faithful? Let X be the set of all subgroups of G=A_4. We define the group action$$G\times X\ni(g,H)\mapsto gHg^{-1}\in X$$I am trying to determine whether this action is faithful, i.e. \bigcap_{H\in X} ... 2answers 112 views ### Product of disjoint cycles question. Consider the following permutations x and y in S_6: x=(1 \, 3 \, 5)(2 \, 4) and y=(2 \, 3 \, 4 \, 5) Express xy as a product of disjoint cycles. My attempt: I first got xy = (3 \, 5 \, ... 2answers 1k views ### Cayley's Theorem - Discourse on Fraleigh's proof or Alternative Easier proof [duplicate] I have hard time understanding Fraleigh's proof. Can someone either explain Fraleigh's or provide an alternative, perhaps easier proof? Thanks so much. 1answer 417 views ### Literature on group theory of Rubik's Cube While searching for literature on the group theory of Rubik's Cube, I mostly find introductions to group theory motivated by applications to Rubik's cube. I.e. the focus lies on elementary group ... 3answers 993 views ### Is there a quick trick to write permutations of S_n as products of transpositions? If I want to write (123) as product of transpositions, I get (13)(12). For (132) I get (12)(13). For (1234), I get (14)(13)(12). Seems like I can write (abcd) as (ad)(ac)(ab). Is this ... 3answers 443 views ### Showing that a transitive abelian permutation group is necessarily regular I am trying to show that a transitive, abelian permutation group acting on a set X is necessarily regular, given this hint: 'Given g \in G, consider the set X^g:=\{x \in X\,|\,gx=x\}. Show that ... 1answer 87 views ### The set that a group can act on it as a permutation group What can we say about the set that a group can act on it as a permutation group? My question is about the structure of elements of a group G when acts on an arbitrary set Ω as a permutation group. For ... 3answers 147 views ### Problem involving permutation matrices from Michael Artin's book. Let p be the permutation (3 4 2 1) of the four indices. The permutation matrix associated with it is$$ P = \begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 ...
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 ...