1
vote
2answers
38 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
29 views

How many distinct elements does a group of permutation on 3 letters have?

I am having some problems solving a problem similar to this. So i tried making it a more simpler problem. I really don't know how to approach this kind of problem. A hint would be very much ...
1
vote
3answers
39 views

Writing a permutation group in 2 row notation

I have a permutation group in $S_7$, namely: $$(12345)(137)(56)$$ How do I write this in two row notation? I am to write it as disjoint cycles and then as transpositions but I feel better working in ...
2
votes
2answers
36 views

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 $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
1
vote
1answer
44 views

Why is this the method to getting transpositions from disjoint cycles?

I have the disjoint cycle: $$(156)(2437).$$ Apparently the "method" would get us: $$(1,6)(1,5)(2,7)(2,3)(2,4).$$ Basically you take the first number, and put it as a transposition of the last number ...
0
votes
1answer
14 views

Prove that cyclic index of this operation can be expressed by formula

Let $T_1$ and $T_2$ be disjoint finite sets and let $G_1$ and $G_2$ be, respectively, some groups of permutations of this sets. Direct sum $G_1 \bigotimes G_2$ acts on $T_1 \cup T_2$: $$ \langle ...
0
votes
1answer
32 views

Showing that there is a permutation $\rho$ that fixes a number that $\sigma$ moves when $\rho \sigma \rho^{-1}=\sigma^{-1}$

Doing an assignment, getting a bit frustrated with this exercise, would really appreciate some help. The first exercise explains what $\sigma$ and $\rho$ are: Let $\sigma$ be the $r$-cycle ...
1
vote
1answer
44 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
2answers
22 views

every k cyclic is a product of at least k-1 distinct tranpositions

There is a theorem says if $A$ in $S_n$ is a $k$ cycle, and $A = a_1 a_2 a_3 \dots a_m$, where $a_i$ are transpositions, then $m \geq k-1$. But how to show there are at least $k-1$ distinct ...
1
vote
1answer
37 views

What mathematical algebra explains sequence of circular shifts on rows and columns of a matrix?

Consider a simple matrix (3X3) with entries thus: [1 2 3; 4 5 6; 7 8 9;] Circular shifts can be performed on any row or any column thus: row-(1/2/3)-(right/left) and column-(1/2/3)-(up/dn) ...
2
votes
2answers
59 views

Find the center of the symmetry group Sn.

Find the center of the symmetry group $S_n$. Attempt: By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\ g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is ...
0
votes
1answer
62 views

How to show that a permutation form a group?

Given the following $12$ permutations: $\{(1), (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(32)\}$ (a) Show that the 12 permutations form a group. (b) Find ...
2
votes
0answers
18 views

Clarification on a cycle parity proof

Prove that cycle $(a_1a_2...a_k)$ is even $\iff$ $k$ is odd. This makes intuitive sense because $(a_1a_2a_3...a_k)=(a_1a_k)(a_1a_{k-1})...(a_1a_3)(a_1a_2)$ which will be an even number of ...
1
vote
2answers
54 views

what is degree of permutation group?

Is "degree" the same term as "order" of a permutation group?
1
vote
2answers
37 views

Permutation help

Consider the elements of $S_7$. For each $\sigma \in S_7$ there is a smallest positive integer |$\sigma$| such that $\sigma^{|\sigma|}=e$. Find the value of $N$= max{ $|\sigma|$ | $\sigma \in S_n$}. ...
0
votes
1answer
26 views

Finding a permutation from a power of itself

Find a permutation $\sigma \in S_9$ such that $\sigma^2=(13579)(268).$ So I know that $\sigma^{10}=\sigma.$ But I don't know $\sigma^5$..... Is $\sigma^{10}=\sigma^4\sigma^6$? I doubt this is the ...
2
votes
1answer
33 views

Signature of permutations is a homomorphism

Given the following definition of $signature$: $\epsilon(\sigma)=(-1)^{n-k}$, where $k$ is the number of cycles (with disjoint supports, counting the 1-cycles) of the permutation, prove that ...
0
votes
1answer
53 views

Odd Permutations

Prove that the product of two odd permutations is even. I'm having a difficult time doing this in the general case. I have that if s is even, then $$\alpha = ...
1
vote
1answer
75 views

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, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
2
votes
0answers
41 views

Symmetry of the pentagon and even permutation

I was doing part (iii). For the first part of that questions, $ D_{10} = \{e, \rho, \rho^2, \rho^2, \rho^4, \sigma\rho, \sigma\rho^2, \sigma\rho^3, \sigma\rho^4 \} $ where $\rho = (1 \ 2 \ 3 \ 4 \ ...
1
vote
0answers
41 views

Finding a permutation $ \alpha $ given $ \alpha^4 $ [duplicate]

I have the following question: Find a permutation $\alpha ∈ S_7 $ such that $\alpha^4 = (2 1 4 3 5 6 7)$. Is $\alpha$ unique? How should I go about this? I've tried a few different trial and error ...
0
votes
1answer
40 views

permutations and transpositions in even and odd cases

Say we had some $\sigma = (1, 2)(2, 3)...(n-1 ,n)$ could someone explain why this formula doesn't hold for odd n? For instance, $n = 2m+1$ $\sigma = (1,2)...(2m-2,2m-1)(2m, 2m+1)$, why does that not ...
2
votes
2answers
31 views

List of all elements of $A_4$ - Jamie Mulholland p. 85

p. 72: $m$-cycle $\iff m - 1$ transpositions. Hence 3-cycle $\iff 2$ transpositions. I condone all the calculations overhead, but I don't understand the proof blueprint. (1.) How do you ...
0
votes
2answers
51 views

Prove this is a subgroup: Subset of $S(A)$ consisting of all the permutations $f(a) = a$

Let $A$ be a set and $a \in A$. Let $G$ be the subset of $S(A)$ consisting of all the permutations $f$ of $A$ such that $f(a)=a$. Prove that $G$ is a subgroup of $S(A)$. I really have no clue how to ...
2
votes
0answers
55 views

Symmetric Group $S_3$

I just wanted to make sure I am thinking about this correctly. I would like to take the product $(123)(231)$. Here, $2-3-1,3-1-2,1-2-3 \Rightarrow (123)(231)=(132)$.
6
votes
1answer
67 views

Why is the parity of a permutation an important concept?

In Pinter's A Book Of Abstract Algebra, the author states that: A number of great theorems of mathematics depend for their proof (at that crucial step when the razor of logic makes its decisive ...
3
votes
2answers
80 views

Example of a simple graph isomorphic to a permutation group.

I'm taking a first course in graph theory this semester and I'm working trough Graph Theory with Applications by J.A. Bonday and U.S.R. Murty. I can't find an answer to question 1.2.12(f): (a) ...
1
vote
1answer
62 views

Permutation that fix elements within set is subgroup?

(a) Let $A$ be a finite set, and $B\subseteq A$. Let $G$ be the subset of $S_A$ (permutations of $A$) consisting of all the permutations $f$ of $A$ such that $f(x)\in B$ for every $x\in B$. Prove ...
5
votes
1answer
108 views

Intuition - Identities with 2-Cycles and 3-Cycles - Mulholland p. 69, 86 - Fraleigh p. 90

Jamie Mulholland p. 69 Theorem 6.1 or Fraleigh p. 90 Corollary 9.12 Any permutation of a finite set of at least two elements is a product of 2-cycles. $1. (a_1, a_2, ···,a_n)= (a_1, a_n)(a_1, ...
2
votes
3answers
76 views

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 ...
0
votes
2answers
65 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
68 views

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, $\langle (12\cdots n),(12)(34)\rangle =\mathcal A_n$. ...
3
votes
1answer
80 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 ...
0
votes
1answer
46 views

How do I prove isomorphism?

I need to prove this: $$S_{\mathbb{N}}\cong S_{\mathbb{Z}}$$ ($S$ means permutation). I'd like to get ideas how to prove it... Thank you!
-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... $\mathbb{Z}_{21}\times \mathbb{Z}_{50}^*\cong\mathbb{Z}_{420}$ ...
-4
votes
1answer
51 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
111 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
0answers
61 views

Question about permutation cycles

We have: $\alpha = (a_1a_2 \cdots a_r), \beta=(b_1b_2\cdots b_r)\in S_n$ ($\alpha,\beta$ are strange cycles) How can we find $f\in S_n$ s.t.: $$\beta=f\alpha f^{-1}\;\;\;?$$ Thank you! (The answer ...
1
vote
2answers
64 views

How to multiply permutations

I didn't understand the rules of multiplying permutations. I'll be glad if you can explain me... For example: we have $f=(135)(27),g=(1254)(68)$. How do I calculate $f\cdot g$?? Thank you!
1
vote
1answer
77 views

Raising a cycle to a power, cycle decomposition

Let $\alpha$ be an m-cycle. Is it true that $\alpha ^k$ can be decomposed into $\gcd(m,k)$ disjoint cycles? for example $(1 2 3 4 5 6)^2 = (1 3 5)(2 4 6)$, $(1 2 3 4 5 6 7 8)^6 =(1 7 5 3)(2 8 6 ...
2
votes
1answer
59 views

How to show that an automorphism of $S_n$ is inner?

Given an automorphism $\phi:S_n\rightarrow S_n$ such that it maps all the transpositions on the transpositions, how do I show that this map is given by a conjugation with an element $s\in S_n$? ...
2
votes
2answers
68 views

Prove that $S_n$ satisfies the following property: if $g \in S_n$, then $g$ and $g^{-1}$ are conjugate in $S_n$.

So I tried setting up an arbitrary $g$ such that $g(a_1)=b_1, ... g(a_k)=b_k$, and then I fizzled out. So I'm showing that for every a in $S_n,\ ag(a^{-1}) = g^{-1}$. I'm showing that they have the ...
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 ...
1
vote
0answers
105 views

Assumption in proof: The alternating group $A_n$ is simple.

Let $N$ be a nontrivial normal subgroup of $A_n$, $n \ge 5$, where $A_n$ denote the alternating group. The book want to prove that $N$ contains a 3-cycle. (Niels Lauritzen, Concrete Abstract ...
2
votes
1answer
105 views

Does a homomorphic image of even permutations consist of even permutations?

If $f:S_n \to S_n$ is a homomorphism, prove $f(A_n) \subseteq A_n$. If every image of a transposition is even, then there is nothing to prove, but it is not sure.. How can I prove the problem?
1
vote
1answer
66 views

Order of a product of two cycles

Is it true or false? If $a$ is a permutation that is an $m$-cycle and $b$ is the permutation that is $n$-cylce then order of $ab = \operatorname{lcm}(m,n)$.
1
vote
5answers
115 views

When is round-robin scheduling possible and with in minimal time?

Suppose that you have six teams $x_0, x_1, x_2, x_3, x_4, x_5$. Can you schedule round-robin games between them so that if one game is played each day, the series of games can be completed in five ...
0
votes
2answers
50 views

Groups of Pemutations

I am having a difficult time with the following question: Find a four element abelian subgroup of S5 and then write its table. I am lost as to where to start. Do I arbitrarily choose 4 elements of ...
0
votes
3answers
153 views

Decomposition of a cycle as a product of transpositions

Can someone please explain the rules pertaining to different ways to write a cycle decomposition as products of 2-cycles, an example from textbook: I understand this $$ (12345) = (54)(53)(52)(51) ...