Tagged Questions
1
vote
1answer
30 views
Computing the order, inverse, and parity of a permutation
How do you compute the order, inverse and parity of $\alpha=(12)(43)(13542)(15)(13)(23)$? Please explain all steps taken to get the answer.
I guess my thought process was to first put it into a ...
3
votes
1answer
45 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 ...
3
votes
4answers
96 views
What is the order of $\tau = (5\ 6\ 7\ 8\ 9)(3\ 4\ 5\ 6)(2\ 3\ 4\ )(1\ 2)$? Is $\tau$ an even or an odd permutation?
In $S_9$, what is the order of $\tau = (5\ 6\ 7\ 8\ 9)(3\ 4\ 5\ 6)(2\ 3\ 4)(1\ 2)$? Is $\tau$ an even or an odd permutation?
For the first question: I tried to write $\tau$ as the composition of ...
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 ...
3
votes
2answers
54 views
Unable to get to all permutations after $n-1$ transpositions
Problem: Give an example of a permutation of the first $n$ natural numbers from which it is impossible to get to the standard permutation $1,2,\ldots,n$ after less than $n-1$ transposition operations ...
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
151 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.
2
votes
2answers
42 views
Why cannot the permutation $f^{-1}(1,2,3,5)f$ be even
Please help me to prove that if $f\in S_6$ be arbiotrary permutation so the permutation $f^{-1}(1,2,3,5)f$ cannot be an even permutation.
I am sure there is a small thing I am missing it. Thank you.
1
vote
2answers
50 views
Notation of permutation
I have a question about the notation of the permutation. I am looking at a proof that shows composition of two permutations is a permutation. There it says, let us assume the contrary, let ...
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 ...
2
votes
1answer
49 views
Number of Permutations Fixed by the Fundamental Transformation is Fibonacci
Writing a permutation in $S_n$ as a product of disjoint cycles, we define a standard representation by writing each cycle with its largest element first, and ordering the cycles by the increasing ...
2
votes
1answer
46 views
Algebra, groups and permutations
The question asks for me to write down the permutations on the set $\{1,2,3,4\}$ which are symmetries of the square with vertices as shown. Hence show that $D_4$ is a subgroup of $S_4$.
1 2
4 ...
3
votes
1answer
91 views
What is the number of even and odd permutations that satisfies the following condition?
Let $\phi$ be a permutation of $n$ numbers with $\phi(1)=1$ and $\phi(2)=2$. It is asked to prove that the number of odd permutations of $n$ numbers that commute with $\phi$ is equal to the number of ...
3
votes
2answers
73 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
120 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
82 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 ...
1
vote
2answers
82 views
How do you calculate the number of elements conjugate to a particular permutation?
Working in $S_n$, and given a particular permutation, how would you go about calculating the number of conjugate elements to the permutation?
I guess since two elements are conjugate iff they have ...
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
3answers
187 views
Let $\alpha=(1234)(5876)$ and $\beta=(1537)(2648)$ belong to $S_8$.
Let $\alpha=(1234)(5876)$ and $\beta=(1537)(2648)$ belong to $S_8$. Determine whether there exists a subgroup of $S_8$ that contains $\alpha$ and $\beta$ and is isomorphic to $D_4$.
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 ...
3
votes
2answers
104 views
Suppose that $\alpha$ and $\beta$ in $S_4$, $\alpha (3) = 4$, $\alpha\beta = (3412)$, $\beta\alpha = (3241)$, find $\alpha$ and $\beta$
Suppose that $\alpha,\beta \in S_4, \,\alpha (3) = 4,\, \alpha\beta = (3412), \,\beta\alpha = (3241).\;$ Find $\alpha\,$ and $\, \beta$.
My attempt:
Ok so I know
$\alpha(3) = 4$ and ...
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 ...
3
votes
3answers
134 views
Is there a way to get all the permutations of $S_4$
I need to calculate the determinant of a $4 \times 4$ matrix by "direct computation", so I thought that means using the formula
$$\sum_{\sigma \in S_4} (-1)^{\sigma}a_{1\sigma(1)}\ldots ...
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 ...
5
votes
1answer
97 views
Do permutation isomorphic actions produce isomorphic semidirect products?
Let $H$ and $K$ be two groups and suppose
$H$ acts on $K$.
$H$ preserves the group structure of $K$.
$\phi$ is the permutation representation of the action:
$\phi:H\rightarrow Aut(K)$
...
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 ...
3
votes
2answers
219 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, ...
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 ...
5
votes
3answers
108 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 ...
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
79 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 ...
