Tagged Questions

1
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1answer
27 views

Symmetry of Plancherel measure (for $S_n$)

For each $n \geq 1$ consider the reverse lexicographical order on the set $P(n)$ of partitions of $n$. Example for $n=7$: $$ \begin{pmatrix} \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 ...
3
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2answers
74 views

How to enumerate subgroups of each order of $S_4$ by hand

I would like to count subgroups of each order (2, 3, 4, 6, 8, 12) of $S_4$, and, hopefully, convince others that I counted them correctly. In order to do this by hand in the term exam, I need a ...
0
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2answers
38 views

When does $S_n$ have a subgroup with order $p^2$ where $p$ is prime?

I'm attempting this homework problem, and I'm not sure where to start. Here is the problem and how what I've got so far. Let $p$ be a prime number. What is the least positive integer $n$ such that ...
1
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2answers
48 views

Conjugation in $S_n$

We have that any element in $S_n$ is generated by the adjacent transpositions $(12),\dots,(n-1,n)$. I am trying to calculate the conjugation of $(ab)\in S_n$ by $\sigma\in S_n$. So if we write sigma ...
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 $\lambda_x (g) = xg$. ...
0
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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
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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.
0
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1answer
50 views

Let $\alpha$ be a $2$-cycle and $\beta$ be a $t$-cycle in $S_n$. Prove that $\alpha\beta\alpha$ is a $t$-cycle

$\bf Claim:$ Let $\alpha$ be a $2$-cycle and $\beta$ be a $t$-cycle in $S_n$. Prove that $\alpha\beta\alpha$ is a $t$-cycle. If $\alpha$ and $\beta$ are disjoint, they commute and thus the product ...
1
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2answers
65 views

Efficient method to determine the order of a permutation in $S_n$

Instead of trying multiplication again and again until I get $(1)(2)(3)(4)(5)(6)(7),$ is there an efficient, logical method to compute order of $(157)(134)(12)$ of $S_{10}$? Is there some relation to ...
4
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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 ...
0
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1answer
88 views

Why must a finite symmetry group be discrete?

I'm having trouble justifying why a finite symmetry group is discrete. Can someone help?
4
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0answers
49 views

What is the centralizer of the Young symmetrizer?

I have read a lot about idempotents, several important facts were about central idempotents. Now, the Young symmetrizer is a constant away from an idempotent, but I don't think it's central. ...
4
votes
2answers
164 views

On Symmetric Group $S_n$ and Isomorphism

I use Abstract Algebra by Dummit and Foote to study abstract algebra! At page 120, section 2 in chapter 4, there is a great result form my point of view which proves that, for any group $G$ of order ...
9
votes
1answer
134 views

Alternating group $A_n$ where $n\geq 5$

I tried so much to prove the following fact about the alternating groups $A_n$, $n\geq 5$ but I couldn't prove it. Any answer or hint will be appreciate; Any maximal ...
1
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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 ...
2
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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 ...
1
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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 ...
1
vote
1answer
71 views

Let $P$ be a Sylow $p$-subgroup of $\operatorname{Sym}(n)$. If $p$ does not divide $n$, then $P\leq\operatorname{Sym}(n-1)$

My question is at the title. I can compute the order of $P$ and see that there exists a Sylow $p$-subgroup $Q$ of $\operatorname{Sym}(n-1)$ which is order $|P|$. But how can we say that $P$ can be ...
1
vote
1answer
85 views

Why is $(1,2,…,p)$ in the center of a Sylow $p$-subgroup of $S_n$?

Assuming $p$ divides $n$, let $P$ be a Sylow $p$-subgroup of $S_n$ and let $z=(1,2,...,p)$. Why is $z$ in the center of $P$? Thanks!
0
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3answers
72 views

A step in the proof of Cauchy's theorem for groups

Let $G$ a group, $p$ a prime and $X=G^p$. Let $\sigma\in S_X$ act as follows: $\sigma(x_1,...,x_p) = (x_2,...,x_p,x_1)$. Let $Y$ the subset of elements in $X$ such that $x_1x_2...x_p=1$ and let ...
0
votes
1answer
95 views

Injective Homomorphism on $S_n$

I have the following question on a homomorphism between symmetric groups and it has been really a pain. I know I am supposed to use induction but, I seem to miss something essential so if anybody ...
3
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1answer
54 views

Using the 6 sets of sets of 5 disjoint 2-cycles and the 3 sets of 5 pairwise disjoint triangles to exhibit an outer automorphism of $S_6$.

As a follow-up to the construction of a graph that $S_6$ acts on in Constructing a graph based on numbers of vertices, incident edges, and incident triangles, I am now specifically looking at the ...
4
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2answers
54 views

$P\in \operatorname{Syl}_p(S_n)$ implies that $P\in \operatorname{Syl}_p(A_n)$ and $|N_{A_n}(P)|=\frac{1}{2}|N_{S_n}(P)|$

$\newcommand{\Syl}{\operatorname{Syl}}$ This is an exercise (with hint about the second part) in my own language book in Group theory, however, maybe it is a lemma or theorem in an standard book ...
7
votes
2answers
210 views

Order of a set $X$ acted upon transitively by the Symmetric Group

Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$. Show that either $|X| \le 2$ or $|X| \ge n$. Small steps ...
1
vote
1answer
163 views

normalizer of a p-Sylow on $S_p$

Let $P$ be a group of order p, on $S_p$ , How can I prove that the cardinality of normalizer of $P$ it's $p(p-1)$ ? If I compute that the number of conjugates of the group P, it's $ \frac{{n!}} ...
5
votes
1answer
104 views

$|G|=12$ and it is isomorphic to $A_4$?

During reading a book, I have faced to this problem telling: $G$ is a group of order $12$ such that $Z(G)$ has no element of order $2$ . Then $G≅A_4$. Obviously, this group is not abelian and I ...
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 ...
2
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2answers
140 views

Symmetry group - symmetry types

I am reading about symmetries in Wikipedia and I am trying to understand this statement: Two geometric figures are considered to be of the same symmetry type if their symmetry groups are ...
2
votes
1answer
308 views

Symmetric group and commuting elements

What is the normalizer of the subgroup $\langle(1,2,...,p)\rangle$ in $S_p$? Clearly it will have to contain the subgroup $\langle(1,2,...,p)\rangle$, but also some additional $p-1$ elements that ...
2
votes
2answers
131 views

$A_5$ problem with normal subgroup

Let G=$A_5$ and $H=\bigl\langle (12)(34),(13)(24)\bigr\rangle$. Prove $(123) \in N_{G}(H)$ and hence deduce the order of $N_{G}(H)$. I know you claim that $A_5$ is simple, then $N_{G}(H)$ has ...
4
votes
2answers
234 views

What are the least sets of generators for $S_n$

Nearly all the books I read give $S_n$ $(n \geq 2)$ and the generating set $\{(i,i+1) | 1 \leq i < n \}$ as an example when talking about presentation groups. But is $\{(i,i+1) | 1 \leq i < n ...
8
votes
1answer
621 views

Enumerating all subgroups of the symmetric group

Is there an efficient way to enumerate the unique subgroups of the symmetric group? Naïvely, for the symmetric group $S_n$ of order $\left | S_n \right | = n!$, there are $2^{n!}$ subsets of the group ...
2
votes
1answer
193 views

Normal subgroups of $S_N$

Is there a list of all normal subgroups for $S_N$? What is a criteria for a finite group to be a normal subgroup of $S_N$? Which of them are kernels of irreducible representation? From a partition ...
3
votes
2answers
338 views

Explanation of claim in Dummit and Foote

Dummit and Foote, p. 204 They suppose that $G$ is simple with a subgroup of index $k = p$ or $p+1$ (for a prime $p$), and embed $G$ into $S_k$ by the action on the cosets of the subgroup. Then they ...
9
votes
3answers
610 views

Generalization of index 2 subgroups are normal

Let $G$ be a finite group and $H$ a subgroup of index $p$, where $p$ is a prime. If $\operatorname{gcd}(|H|, p-1)=1$, then $H$ must be normal. Does somebody have a quick proof of this?