Tagged Questions

4
votes
1answer
47 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
2answers
87 views

Can you find an isomorphic group?

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
109 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
89 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
126 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
101 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
68 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
80 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
114 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
41 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
55 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
51 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
152 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.
5
votes
1answer
95 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
93 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
2answers
75 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
39 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
121 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 ...
4
votes
2answers
83 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
36 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
80 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 ...
2
votes
5answers
84 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
111 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
178 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
66 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
56 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
55 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
68 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 ...
2
votes
1answer
75 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 ...
22
votes
1answer
433 views

Six Frogs - Puzzle

I had come across a puzzle: The six educated frogs in the illustration are trained to reverse their order, so that their numbers shall read 6, 5, 4, 3, 2, 1, with the blank square in its ...
2
votes
1answer
78 views

I want to find a natural faithful action with the wreath product.

Let $A$ and $B$ be any sets. Let $G\leq \operatorname{Sym}(A)$ and $H\leq\operatorname{Sym}(B)$. Can any one find a faithful action of the wreath product $G\wr H$ on $A\times B$? If there is no such ...
3
votes
0answers
63 views

Is there a simple way to show that wreath product is associative?

Is there a simple way to show that wreath product is associative? If your proof is short, please write it explicitly. Thank you.
1
vote
1answer
93 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 ...
6
votes
1answer
300 views

When does the adjacency or incidence matrix of a graph have consecutive ones property?

Given a graph, what are some sufficient (and necessary) conditions to tell if its adjacency matrix has the consecutive ones property? Similar question for its incidence matrix? Note that a ...
1
vote
1answer
59 views

A group that has a $\frac{3}{2}$-transitive subgroup

Do you know a group that has a $\frac{3}{2}$-transitive subgroup and it is not $\frac{3}{2}$-transitive itself?
3
votes
1answer
111 views

A faster way to show that a subgroup is normal

I'm working with $\mathbb S_4$, and I have a subgroup of $\mathbb S_4$ called $G$. $G$ is generated by $a=(12)(34)$ and $b=(123)$, which I've actually found to be $A_4$ by multiplying elements by $a$ ...
2
votes
3answers
80 views

$S_n$ acting transitively on $\{1, 2, \dots, n\}$

I am reading Dummit and Foote, and in Section 4.1: Group Actions and Permutation Representations they give the following example of a group action: The symmetric group $G = S_n$ acts transitively ...
1
vote
3answers
85 views

Prove that $S_n$ is doubly transitive on $\{1, 2,…, n\}$ for all $n \geqslant 2$.

Prove that $S_n$ is doubly transitive on $\{1, 2,\ldots, n\}$ for all $n \geqslant 2$. I understand that transitive implies only one orbit, but...
2
votes
3answers
264 views

Show group of order $4n + 2$ has a subgroup of index 2.

Let $n$ be a positive integer. Show that any group of order $4n + 2$ has a subgroup of index 2. (Hint: Use left regular representation and Cauchy's Theorem to get an odd permutation.) I can easily ...
1
vote
1answer
79 views

Help needed in proving a theorem on a permutation that is the product of dis. cycles of prime length.

So we were given the following proof to do: Let $ p $ and $ q $ be distinct primes. Suppose $ \alpha $ is a permutation of $ S_n $ and suppose $ \alpha = \gamma_1 \gamma_2 $ where $ \gamma_1 $ and $ ...
1
vote
0answers
94 views

Primitivity implies transitivity?

I am noting a simple problem about a permutation group from "Permutation Group" By J.Dixon, its answer and my attempt to understand it in details: Q: A primitive permutation group $G(≠1$) is ...
2
votes
1answer
61 views

Is there any permutation $x≠1$ leaving at least $n-2k$ letters fixed at this group?

This question has an answer which I am noting both here. Q: Suppose that $G$ is permutation group of degree $n$. If for an integer $k$ where $4≤2k≤n$ we have $|G|≥(n-k)!k$ then $G$ contains a ...
3
votes
1answer
103 views

If $G$ is a transitive permutation group then $\mathrm{fix}(G_\alpha)$ is a block

I am new here and don't know much about Latex so, I attach my question from Permutation Groups by J. Dixon. I hope to get a help for it: 1.6.5 Let $G$ be a transitive subgroup of ...
1
vote
1answer
199 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 ...
3
votes
1answer
134 views

Find the subgroups of index two of this finite semi-direct product

This older stackoverflow question may be helpful in answering the question that I ask below, although I could not work it out. For $n\geq 1$, let $X=\lbrace 1,2, \ldots ,n \rbrace$, $Y=X \cup (-X)$ ...
2
votes
2answers
247 views

How does one decompose the regular representation of $S_3$?

I need to decompose the regular representation of $S_3$ into irreducible ones. What I know so far is this: $S_3$ is generated by $\tau = (12)$ and $\sigma = (123)$. If $v$ is an eigenvector of ...
4
votes
1answer
87 views

Is there a “natural” transitive action of $SL_2(\mathbb{F}_5)$ on a set with 5 elements?

I'm really looking for a "cute" way of showing that $SL_2(\mathbb{F}_5)$ is a double cover of $A_5$. The sort of action I am looking for is something like the action of $GL_2(\mathbb{F}_3)$ on ...
0
votes
1answer
181 views

What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters?

What is the number of automorphisms (including identity) for permutation group $S_3$ on 3 letters? I believe the answer for this is 6. As we can write the group elements as below (a)(b)(C) (ab)(c) ...
7
votes
2answers
104 views

Four generators of $S(9)$ - A smart way of showing that this generates the entire group?

For the first time, I'm going to post a group theory question. I have four 4-cycles, given by: $(1452),(2563),(4785),(5896)$. I know that the group generated by these guys are $S(9)$ by brute ...
3
votes
2answers
180 views

Existence of subgroup of order six in $A_4$

Show that the alternating group $A_4$ of all even permutations of $S_4$ does not contain a subgroup of order $6$. For me am thinking to write all elements of $A_4$ and trying to find every ...

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