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.

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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.), ...
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0answers
11 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 ...
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0answers
19 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)= ...
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0answers
39 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
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1answer
34 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 ...
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0answers
34 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 ...
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0answers
27 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
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1answer
20 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
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1answer
45 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
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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 ...
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2answers
21 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 ...
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3answers
56 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
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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
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0answers
58 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
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1answer
21 views

Minimal expressions of generators of a symmetric group.

${S}_n$ is the symmetric group of $n$ symbols. Let $G_n = \{\ s_n,\ t_n,\ id\ \}$ be the set of generators of ${S}_n$ where $$s_n = (1\ n)$$ $$t_n = (1\ 2\ \dots\ n)$$ $$id\text{ = identity}$$ A ...
3
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0answers
28 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 ...
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0answers
45 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 ...
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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
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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 ...
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1answer
28 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
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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
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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
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0answers
45 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, ...
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3answers
36 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
48 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
70 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 ...
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0answers
42 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
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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 ...
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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 ...
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1answer
48 views

Dihedral groups [closed]

Consider the Dihedral group $G=D_{12}=\langle a,b\rangle$.Which of the following is false? A) $G$ has an element of order $3$. B) All subgroups of $G$ of order $4$ are isomorphic. C) All subgroups of ...
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0answers
19 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 ...
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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
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1answer
57 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
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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
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3answers
39 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
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0answers
53 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 ...
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0answers
76 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
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1answer
36 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
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2answers
47 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 ...
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0answers
33 views

The irreducible representation of $S_n$ of degree $n-1$ [duplicate]

So I understand it's not hard to show that the standard $(n-1)$-dimensional irreducible representation of $S_n$ is the only irreducible representation of $S_n$ of degree $n-1$ using characters/Young ...
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1answer
42 views

symmetric group as a subgroup of general linear group

Is the symmetric group $S_n$ a normal subgroup of the general linear group $GL(n,\mathbb{R})$? We regard $\sigma\in S_n$ acts on $\mathbb{R}^n$ by permuting the coordinates ...
3
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1answer
39 views

Find the order of a subgroup of $S_5$ generated by two elements

The Question I am trying to solve reads : Find the order of a subgroup of $S_5$ generated by the elements $s=(123)$ and $r=(12345)$. Is this group sovable ? Is it nilpotent ? I tried to compute a ...
0
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2answers
39 views

Proof that if $\sigma \in S_n,$ then $\sigma^n = \iota.$

I think I remember my abstract algebra professor mentioning in class that if $\sigma$ is any permutation belonging to the symmetric group $S_n,$ then $\sigma^n = \iota,$ the identity permutation. We ...
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0answers
13 views

Bounding the number of contingency tables

Suppose I am given $c\in\{1,\ldots,N\}^d$ and I want to know, how many non-negative matrices $M\in\mathbf{N}^{d\times d}$ one can find such that both the row and column sums are equal to $c$. I am ...
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0answers
15 views

Restriction of a Specht module to the alternating group

Let $n\in\mathbf{N}$ and denote by $S_n$ the symmetric group on $n$ letters. For $\lambda\vdash n$ a partition of $n$ the Specht module $S^\lambda$ defines an irreducible representation. What ...
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0answers
34 views

View a group acting faithfully and transitively on a set $X$ as a subgroup of a wreath product.

I'm studying algebra and I saw that given $H$ group of permutations of a set $\Delta$ and $K$ group of permutation of a set $\Omega$ we have that the wreath product $H\wr K$ is a group of permutations ...
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2answers
42 views

Every Two Element in A Coset

For every $a,b \in G$, $a,b \in cH$ for some maximal subgroup $H$ of $G$ and some $c \in G$. For what groups is the following property true? I know its true for $\mathbb{Z_m} \times ...
0
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1answer
38 views

proving a matrix is symmetric

Let $A$ be an $nxn $ symmetric matrix. a) Show that $A^2 $ is symmetric. b) Show that $2A^2 -3A + I$ is symmetric. for part a), i have: $A=A^T$ $A^2 = A\times A$ $A^2 = (A^T)\times(A^T)$ $A^2 = ...
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0answers
16 views

Maximal Kostka Numbers

Let $\lambda\vdash n$ be a partition of $n$ and assume that $\lambda$ has $k$ parts. Then let $\mu$ run through all the other partitions of $n$ and consider the Kostka-number $K_{\lambda,\mu}$. Can ...
2
votes
3answers
414 views

Proof that S3 isomorphic to D3*

So I'm asked to prove that $$S_{3}\cong D_{3}^{*}$$ and I know how to exhibit the isomorphism and verify every one of the $6^{2}$ pairs, but that seems so long and tedious, I'm not sure my fingers can ...