1
vote
2answers
46 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
45 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
0answers
30 views

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} ?
2
votes
2answers
35 views

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, ...
0
votes
1answer
34 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 ...
1
vote
2answers
62 views

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). ...
3
votes
2answers
123 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 ...
0
votes
2answers
66 views

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: $(123)$, ...
3
votes
1answer
81 views

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 ...
-6
votes
2answers
50 views

Which of those are isomorphism [closed]

I have here a list. I need to prove which one of them is true (or not) and prove it... $\mathbb{Z}_{21}\times \mathbb{Z}_{50}^*\cong\mathbb{Z}_{420}$ ...
2
votes
2answers
76 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.
-4
votes
1answer
52 views

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...
0
votes
1answer
40 views

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 ...
0
votes
1answer
112 views

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 ...
0
votes
1answer
71 views

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 ...
0
votes
1answer
98 views

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 ...
0
votes
1answer
97 views

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 ...
2
votes
2answers
214 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) ...
1
vote
3answers
194 views

Counting solutions of $x^{30}=1$ in $S_6$

I was asked to count the number of solution of $$x^{30} =1$$ with $x \in S_{6}$ (the group of the permutations over a set with $6$ elements). I started noticing that the divisors of $30$ are: ...
0
votes
1answer
65 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. ...
3
votes
1answer
75 views

Action of $S_7$ on the set of $3$-subsets of $\Omega$

Reviewing the great book in Permutation Groups by J.D.Dixon, I encountered the following problem: Show that $S_7$ acting on the set of $3$-subsets of $\Omega=\{1,2,3,4,5,6,7\}$ has degree $35$ and ...
1
vote
4answers
69 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?
1
vote
2answers
142 views

Permutation representation of group described by $a_i^2=\theta^2=1, a_ia_{i+1}=\theta a_{i+1}a_i=a_{i+2}$.

Let $G$ be a group with elements $\{e, a, b, c, \theta, \theta a, \theta b, \theta c \}$ where $a^2 = b^2 = c^2 = \theta$, $\theta^2 = e$, $ab = \theta b a = c$, $bc = \theta c b = a$, $ca = \theta a ...
7
votes
1answer
344 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 $\lambda_x (g) = xg$. ...
7
votes
1answer
194 views

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 ...
2
votes
2answers
242 views

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 ...
4
votes
1answer
156 views

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?
3
votes
1answer
137 views

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 ...
1
vote
1answer
144 views

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 ...
5
votes
1answer
448 views

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 $p=5$)?
2
votes
3answers
49 views

Computing $\langle (13746) \rangle$ in $S_7$.

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 ...
3
votes
1answer
84 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} ...
0
votes
2answers
93 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 \, ...
0
votes
2answers
1k views

Proof of Cayley's theorem. [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.
7
votes
1answer
341 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 ...
4
votes
3answers
783 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 ...
3
votes
3answers
365 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 ...
1
vote
1answer
78 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 ...
7
votes
3answers
137 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 ...
5
votes
2answers
210 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
1answer
53 views

For which values of $n \ge 2$ is $H_n = \{\alpha \in S_n:|\alpha| \text{ is odd}\}$ a subgroup of $S_n$?

Let $H_n = \{\alpha \in S_n:|\alpha| \text{ is odd}\}$. For which values of $n \ge 2$ is $H_n$ a subgroup of $S_n$? Ok so I figure that since the order is odd, then $\alpha$ can be written as a ...
2
votes
1answer
108 views

Questions about products of $p$-cycles.

Let $p$ be a prime and let $n$ be an integer such that $n \le p$. a) Let $\alpha$ be a p-cycle in $S_n$. Prove that $\alpha^j$ is a p-cycle for $1 \le j \le p − 1$. b) Assume that $2p \le ...
1
vote
5answers
107 views

Permutation question. $qpq^{-1}$, $q,p,r,s \in S_{8}$.

Let $p,r,s,q \in S_{8}$ be the permutation given by the following products of cycles: $$p=(1,4,3,8,2)(1,2)(1,5)$$ $$q=(1,2,3)(4,5,6,8)$$ $$r=(1,2,3,8,7,4,3)(5,6)$$ $$s=(1,3,4)(2,3,5,7)(1,8,4,6)$$ ...
2
votes
2answers
147 views

Express $\alpha^{83} $ as a product of disjoint cycles

I have $\alpha$ = $(15)(37964)(8)(2)$ and am asked to express it to the power of $83$ This is what I have done so far, $\alpha ^{83} = (15)^1(37964)^3(8)(2) \: = (51)(46937) $ Am I doing it ...
1
vote
5answers
2k views

Expression as a product of disjoint cycles

Let $\alpha = (9312)(496)(37215) \in S_n, n \ge 9$. Express $\alpha$ as a product of disjoint cycles. I know this is probably a really easy question, but my professor didn't elaborate on how to ...
1
vote
2answers
95 views

Finding the number of orbits

How many orbits are there of $(12)(25)$ in $S_{5}$? Considering the permutation $(12)$, it has $4$ orbits and is as follows: $\{\{1,2\},\{3\},\{4\},\{5\}\}$ and (25) also has 4 orbits and is also ...
2
votes
1answer
61 views

Sorting numbers in a matrix by moving an empty entry through other entries is not always possible .

let $\mathbf{A}$ be the set of all $n\times n$ matrices on $\mathbb{N}_{n^2}=\{1,2,...,n^2\}$ with distinct entries. let $T$ be the set of all permutations of $\mathbf{A}$ which swap the entry 1 with ...
1
vote
2answers
70 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 ...
5
votes
1answer
84 views

Is it possible to reverse this sequence of permutations?

Let $ S = (a_1, a_2, ..., a_N) $ be a finite (arbitrarily long) sequence of elements, and let $p_1, p_2, ..., p_n $ be the first $n$ prime numbers, with $n \ge 3$. We apply a sequence of ...
3
votes
1answer
185 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 ...