Should be used with the (group-theory) tag. Symmetric group is a group consisting of all permutations of given finite set with composition as the binary operation.

learn more… | top users | synonyms

0
votes
1answer
31 views

proving a group

Consider the group ($[0; 2\pi)$;$\oplus_{2\pi}$) where $\oplus_{2\pi}$ means addition modulo $2\pi$. Define $S_1$ as the following subset of $\mathbb{C}$ $S_1$ = {$z \in \mathbb{C}; z = e^{i\theta}; ...
1
vote
1answer
34 views

Proof of the fact that the set of (p,q) shuffles is a cross section of the subgroup $S_p\times S_q$

Definition Let $G$ be a group and $H$ its subgroup. We name a subset $K$ of $G$ a cross section if it has exactly one element from each left coset of $G/H$. Definition Let $n=p+q$ for some ...
0
votes
0answers
31 views

If $H<S_n$, $H$ is abelian and transitive on $\{1,2,…,n\}$, then the order of $H$ is $n$ [duplicate]

If $H<S_n$, if $H$ is Abelian and transitive on $\{1,2,...,n\}$, then the order of $H$ is $n$. So far I have: $H$ is transitive therefore the group orbit is the group itself. I know I should ...
0
votes
0answers
62 views

The structure of the Sylow p-subgroups of $Sym(n)$

For the structure of the Sylow $p$-subgroups of $Sym(n)$, there is a standard proof by using the properties of the wreath product like as in the Passman's book. But I want to understand this proof. In ...
0
votes
1answer
31 views

Non-simplifiable permutation matrices

The permutation matrices for 2 and 3 dimensions look like this: 2-dimensional: $$\quad M_1^{2d}=\left(\begin{matrix}1 &0\\0 &1\end{matrix}\right), M_2^{2d}=\left(\begin{matrix}0 &1\\1 ...
2
votes
1answer
34 views

Show $A_6$ is simple

I have to show that the group $A_6$ is simple. For the subgroups that have order divisible by $5$ and order of $8,9,18,24,36,$ and $72$ I have shown that those subgroups are not normal. Now we need to ...
1
vote
3answers
19 views

Showing product of disjoint cycle

I am trying to show the product of two disjoint cycles such that they have nothing in common for $A_n$ for $n\ge 3$. So I have the two cycles $(ab)(cd)$. I have read here: ...
4
votes
0answers
145 views
+100

Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
0
votes
0answers
16 views

two books on symmetric groups

Listed below are two books on new approaches to the representation of symmetric groups. Could anyone explain to me the difference/connection between these two approaches? which approach will be more ...
0
votes
1answer
29 views

Combinatorial puzzle concerning labelled equilateral triangles

Consider equilateral triangles $\Delta$ of fixed size and in a fixed position with each side labelled by a label $l \in \{1,\dots,k\}$. Obviously there are $k^3$ such labelled triangles. Let ...
4
votes
1answer
61 views

Flattening Young Tableaux

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_k)$ be a partition with $|\lambda|=n$ and $\lambda_1\geq \lambda_2\geq\cdots\geq \lambda_k$. For any Standard Young Tableaux (SYT) $T$ of shape ...
1
vote
1answer
42 views

When p is prime number, explain all the elements of $S_p$, whose the order is not divisible by any prime numbers less than p

I'm studying a first course in abstract algebra, and currently I'm stuck with this problem. The problem is that, when p is prime number, explain all the elements of $S_p$, whose the order is not ...
3
votes
3answers
127 views

Proof that $A_n$ the only subgroup of $ S_n$ index $2$.

I have what seems to me a very simple proof that $A_{n}$ is the only subgroup of $S_{n}$ of index 2. Since I've seen other people prove it with what feel like really complicated methods (Like here.), ...
0
votes
0answers
14 views

Conjugation of $A_5$ by $S_5$ and effects on irreducible representations of $A_5$

I want to ask q5 of the following https://www.dpmms.cam.ac.uk/study/II/RepresentationTheory/2010-2011/repex2.pdf The group $S_5$ acts on $A_5$ by conjugation. How does this action act on the ...
0
votes
0answers
26 views

permutation representation of Symmetric group

The symmetric group $S_n$ acts on $\mathbb{C}^n$ by permuting the coordinates. Decompose this representation explicitly into irreducible representations. For the action $\sigma (a_1,\cdots, a_n)= ...
0
votes
0answers
42 views

Prove that $S_n$ is a group with respect to composition

Prove that $S_n$ is a group with respect to composition $fn = f\text{ follow }n$. I know that to prove something is a group I have to show identity, inverse, close, and associativity. But I do ...
0
votes
1answer
36 views

Factor Group, Isomorphism

If group $A=S_3⊕\mathbb{Z}_4$ and subgroup $B=\langle (132),2\rangle$, find a group the factor group $A/B$ is isomorphic to and construct the group table for $A/B$. I'm really not sure what to do with ...
0
votes
0answers
35 views

In a group of symmetries, find elements such that $\sigma \circ\beta = \beta \circ\sigma$ and $\sigma \circ\beta = \beta^{-1} \circ\sigma$

(i) Let $\sigma = (12)(345) \in S_5$. Find all $\beta \in S_5$ such that $\sigma \circ\beta = \beta \circ\sigma$. $S_5$ is the symmetric group of 5 elements. (ii) Let $\sigma \in S_n$. Show that ...
1
vote
0answers
28 views

Irreducible representations of $S_3$

I am trying to do a problem from artin's algebra 2nd ed (Chapter 10, Exercise 2.3) but having trouble: Let $(\rho , V)$ be a representation of the symmetric group $S_3$. Let $x=(123), y=(12)$ be the ...
0
votes
1answer
24 views

subgroups with cycles

Let Sym(n) denote the symmetric group on n letters and let H be a subgroup of Sym(n) . Suppose that H contains a k cycle for each value of k from 2 through n . This should be enough to ...
3
votes
1answer
48 views

Representation of the symmetry group (rotations) of the icosahedron

Suppose $I$ is the set of vertices of the regular icosahedron, here is a link of the icosahedron: http://www.werheit.mynetcologne.de/icosaeder.gif Let $F(I)$ be the space of complex functions on $I$, ...
3
votes
1answer
24 views

For $H$ fixed point free show $|H| \leq n$

Suppose that $H \leq S_n$, and suppose that $H$ has the property that all non-identity elements of $H$ are fixed-point free. Show that $|H| \leq n$. I am trying to prove this by induction. For ...
0
votes
2answers
22 views

Problem on symmetric group of order $4$

Let $S_4$ denote the group of permutations of $\{1,2,3,4\}$ and let $H$ be a subgroup of $S_4$ of order $6$ Show that there exists an element $i$ in $\{1,2,3,4\}$ which is fixed by each element of ...
1
vote
3answers
61 views

What is the order of the alternating group $A_4$?

When I write out all the elements of $S_4$, I count only 11 transpositions. But in my text, the order of $A_4$ is $12$. What am I missing? ...
0
votes
1answer
32 views

Elements of order 5 in $A_6$

I am trying to find the elements of order 5 in $A_6$ and I understand that they are of the form $(abcde)$, correct? So the number of elements is $(6*5*4*3*2)/5$=144. I looked somewhere else and it ...
2
votes
0answers
59 views

Show that $G/H\cong S_3$

If $G:=S_4$ and $H:=\{id,(12)(34),(13)(24),(14)(23)\}$ Show that $G/H$ has order $6$ and all of its elements have order less than or equal to $3$ (so by the classification of the groups of order 6 ...
0
votes
1answer
59 views

Words of the Normal Form of the Presentation of a Finite Monoid

Massive Edit: After consulting with a few mathematicians at my university, I got a better understanding of what I was actually looking for. $$ \langle\ s,\ t\ \vert\ s^2 = 1,\ t^n = 1\ \rangle $$ ...
3
votes
0answers
33 views

Cauchy Identity for a specialized product of Schur polynomials

Let $\lambda=(\lambda_1,\lambda_2,\cdots,\lambda_d)$ be a partition, with $|\lambda|=n$. Let $\nu=\nu(\lambda):=(\lambda_1-1,\lambda_2,\cdots,\lambda_d).$ In other words, $\nu$ is obtained from ...
0
votes
0answers
49 views

Permutation and symmetric Group

I have the following task: For all $i,j \in \{1,2,3,4\} $ such that $i+j=5$, let $G$ be the set of permutations $ \sigma \in S_4 $ satisfying $\sigma (i) + \sigma(j)=5$. a.) List all the ...
0
votes
0answers
20 views

Symmetric groups and transitive action

I am trying to show that for $(x,x_1),(y,y_1)$ there exists $g\in S_n$ such that $gx=y$ and $gx_1=y_1$ where $x$, $x_1$, $y$, $y_1\in \{1,2,3,\dots ,n\}$ and $x\neq x_1$, $y\neq y_1$. Is this claim ...
1
vote
1answer
60 views

Find this example

Let $H=\{e,(13)\}$ be a subgroup of $S_3$. Find element $a,b \in S_3$ where $bh_2ah_1 \in aH$ but $bH\ne H$. $h_1$ and $h_2$ are elements in $H$. My friend thinks that it is (123) and (132), but I ...
1
vote
1answer
30 views

Order of a permutation divides n in Sn

Let $\theta \in S_n$, and for any $k \in \mathbb{N}$, either $\theta^k = I_{I(n)}$ or $\theta^k$ has no fixed elements. Show that $o(\theta) | n$. $I_{I(n)}$ denotes the identity. I'm completely ...
2
votes
2answers
21 views

are identities symmetric on both sides of an equation

if the LHS of an identity is symmetric does it mean the RHS must also be symmetric? In addition how do you test if an identity in three variables is symmetric e.g let the three variables be x,y and ...
3
votes
1answer
44 views

Quaternion^2 in the symmetric group

Which is the minimum $n$ such as $Q_8\times Q_8\cong H$ with $H<S_n$? I now that the minimum n such as $Q_8$ embeds in $S_n$ is 8, so $Q_8\times Q_8$ embeds in $S_8\times S_8< S_{16} $, but is ...
2
votes
0answers
48 views

s4≀s2 visually?

What does the polytope whose symmetry is (exactly, not larger isomorphisms) s4≀s2 look like? Does anyone know of a full decomposition of its construction, i.e. lattices, hasse diagrams, cayley graphs, ...
0
votes
3answers
41 views

Computing the inverse of a permutation

I didn't understand the permutation and of course, I got this question wrong. Compute the inverse of the following permutation: $$ \begin{pmatrix} 1&2&3&4&5&6\\ ...
2
votes
1answer
51 views

How to find out if two cycles are conjugate?

Let for example $a=(14395)(26)(78)$ and $b = (154)(2368)(79)$ be elements of $S_9$. I know that by definition, conjugate elements of a group $G$ are elements $x,y \in G$ such that $x=aya^{-1}$ for ...
3
votes
1answer
71 views

Show that the order of $G$ cannot be divisible by $7$.

Show that the order of a proper primitive group of degree $19$ cannot be divisible by $7$. It means as following- This means that a group $G$ which acts on a set $\Omega$ of cardinality $19$ (this ...
0
votes
0answers
43 views

Is every symmetric group generated by some cycles?

Actually, I have two questions here: How to show that $D_3$, the dihedral group, is isomorphic to $S_3$ in details? Is every symmetric group generated by some cycles? For question (1), I know ...
2
votes
1answer
20 views

Verify statement about conjugates in symmetric group

At http://planetmath.org/simplicityofthealternatinggroups it states the following. Let $\pi$ be a permutation written as disjoint cycles \[ \pi = (a_1, a_2, \ldots, a_k)(b_1, b_2, \ldots, b_l)\ldots ...
1
vote
0answers
25 views

Writing out an Alternating Group

I am trying to write out what is in $A_4$ and $A_6$, their general form, not the whole $n!/2$ cause that would be a lot. My main question is how do I do that. I know they are all the even permutations ...
1
vote
0answers
20 views

groups and symmetry

Which of the following may be false? A) Any non-trivial element of $C_n$ generates $C_n$. B) Any subgroup of $C_n$ is cyclic. C) if $m|n$ then $C_n$ has at least one subgroup of order $m$. D) if ...
0
votes
0answers
18 views

I wish to study symmetric groups properly

I wish to study symmetric groups properly. So kindly suggest some good books. Suppose it is available in pdf format kindly send a link..
2
votes
1answer
60 views

Is there any formula to calculate the number of normal subgroups of $S_n$?

Is there any formula to calculate the number of normal subgroups of $S_n$? Suppose i have an answer to this question it is easy to answer how many homomorphism is there from $S_n$ to any other ...
2
votes
2answers
27 views

If $n>m$, then the number of $m$-cycles in $S_n$ is given by $\frac{n(n-1)(n-2)\cdots(n-m+1)}{m}$.

Show that if $n>m$, then the number of $m$-cycles in $S_n$ is given by $$\frac{n(n-1)(n-2)\cdots(n-m+1)}{m}.$$ My doubt Suppose I wish to count the number of $m-$cycles. Then I will get ...
1
vote
3answers
41 views

Alternating groups, specifically $A_6$

My question is what are the possible order of $A_6$? And how would I show I get $\frac{6!}{2}=360$. Any tips? I know that $A_6$ is the group of even permutations on six elements. I also know that ...
3
votes
0answers
54 views

Automorphism on Symmetric Group and Transpositions

I've been looking into the group of automorphisms of the symmetric group $S_{n}$ for when $n > 6$. Something which is claimed frequently is that if an automorphism sends a transposition to a ...
1
vote
0answers
77 views

Is this problem still open or solved?

Problem- Let $n\geq$ and let $T$ be the set of all permutations in $S_n$ of the form $t_k=\prod_{1\leq i\leq k/2}(i,k-i)$ for $k=2,3,4.....(n+1)$. Then find the least integer $f_n$ such that ...
0
votes
1answer
38 views

Proof of isomorphism between $D_{2n}$ and $D_n \times Z_2$ for $n$ odd

If we define a function $\phi : D_{2n} \rightarrow D_n \times Z_2$ for odd $n$ and we want to show that it is an isomorphic function, I am not very sure how to do it. We know that $D_{2n} = \{e, r, ...
2
votes
2answers
56 views

Centre of the group S4

Quite a simple looking question guys, Find the centre $Z(S_4)$ of $S_4$. The previous part asked me to find centralizers for $S_4$. I note that $Id$ is the only element contained in everything so I ...