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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10 views

If $g=(n-j,\dots,n)$ and $\sigma\in S_{n-1}$, why are the inversions of $g\sigma$ the union of the inversions of $\sigma$ and $g$?

I can't see why a claim I'm reading is true. If $\sigma\in S_n$, let $R(\sigma)=\{(i,j):i<j,\ \sigma(i)>\sigma(j)\}$, i.e., $R(\sigma)$ is the set of inversions of $\sigma$. The set ...
4
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4answers
133 views

Is there any neat way to show $\phi$ is a homomorphism?

In Michael Artin's Algebra (chapter 2, page 50, example 2.5.13) the author illustrates a homomorphism from $S_4$ (all permutations of indices $(1,2,3,4)$) to $S_3$ (all permutations of indices ...
6
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1answer
60 views

Norm $\Vert \cdot \Vert$ on the symmetric group $S_n$

If we define a real valued function $\Vert \cdot \Vert$ on the $n^{th}$ order symmetric group $S_n$ satisfying following conditions $$\begin{align} & \|x\|=0\iff x=\omega\,\,\,(\text{identity ...
1
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2answers
45 views

How to solve conjugation equations in group theory

Given the permutations $(12)(34)$ and $(56)(13)$ find $a$ such that $$a^{-1}xa = y$$ I just realized that I don't know how to solve this exercise. My book don't even give examples of how to solve ...
2
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1answer
50 views

Counting number of bijections

The question is: Let $S = \{a,b,c,d\}$ and let $X : = \{f\colon S \to S \mid f \, \, \text{is bijective and } f(x) \ne x \, \, \text{for each}\, \, x \in S \}$. What is $|X|$? Is there a simple ...
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2answers
54 views

If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
1
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1answer
34 views

Presentation of the symmetric group of 5 symbols.

I am trying to write the presentation of the symmetric group $S_{5}$. We know that $S_{5}$ is generated by $a=(1,2)$ and $b=(1,2,3,4,5)$. Using this I am trying to write presentation of $S_{5}$. My ...
1
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0answers
36 views

Possible orders of the elements of the alternating group $A_n$

What are the possible orders of the elements of the alternating group $A_n$?
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0answers
13 views

crystallographic space group

Where can I find a table showing all the space group operations in matrix form for all crystals? Are there any restrictions on the translation part of the space group operator should be for a crystal? ...
3
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1answer
33 views

Representation-theoretical reasons for positivity of product of two Schubert polynomials?

In the Wikipedia article on Schubert polynomials there is a claim that there are representation-theoretical reasons for the product of two Schubert polynomials to have nonnegative coefficients when ...
3
votes
1answer
30 views

Which permutation am I? Or: what is a bijection $f:S_n \rightarrow \{1,2,\ldots,n!\}$ such that we can compute $f(\beta)$ easily?

Let $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$ and assume that $S_n$ is ordered in some way, i.e., $$S_n=\{\alpha_1,\alpha_2,\ldots,\alpha_{n!}\}.$$ We are able to choose this ordering on ...
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2answers
112 views

More Symmetric than the symmetric groups?

So I was considering the following question. Is there a group of size n! (or less), that contains a larger number of subgroups than $$S_n$$? In thinking about this problem one obvious contender that ...
11
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2answers
104 views

Subgroups of $S_n$ with exactly one fixed point for each element all have the same fixed point.

Let $G$ be a subgroup of $S_n$ (where $n$ is a positive integer) such that each non identity element $g\in G$ has exactly one fixed point. Prove there is an element of $[n]$ that is fixed by every ...
2
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2answers
17 views

proving Caley's Theorm using the conjugacy operation instead of left multiplication

I need to prove Caley's theorem that each group G is isomorphic to a sub group of S(n). Wherever I check it is proven using the operation of multiplying from left side. that means $f_g(x) = g*x$ I ...
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2answers
69 views

Set of generators for $A_n$, the alternating group.

The problem is this: Prove that $A_n = \langle (123),(124),\ldots,(12n)\rangle$. I had cogitated this problem for quite awhile, and haven't been able to come up with anything. The only good idea ...
3
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1answer
58 views

There is no injective morphism from $\mathbb S_7$ to $\mathbb A_8$

I am trying to show that it doesn't exist an injective group morphism $f:\mathbb S_7 \to \mathbb A_8$. If there is an injective morphism from $\mathbb S_7$to $\mathbb A_8$ then $\mathbb S_7$ is ...
2
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2answers
60 views

Prove or disprove $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$

I need to prove or disprove: To the group $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$ My attempt: I just wrote all the details that I know: element in $A_5$ should be in form like ...
0
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0answers
29 views

Maximum order of an element in Alternating group of degree 10.

What is the maximum order of any element in $A_{10}$? My attempt: I tried this problem. But I am not sure about the answer. My answer is 21 because 10 can be written as 7+3. $A_{10}$ can have ...
1
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1answer
36 views

What does it mean to find elements in $S_9$ that are “not cycles”?

I came across this wording in the following question. Some clarification on what this means and how to approach this problem would be helpful. Thanks!
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2answers
43 views

Help with some simpler symmetric group $S_n$ problems.

I apologize if the problems seem trivial but I have not been able to find example problems or solutions to some of these questions. Could someone please confirm my attempts are correct or not? ...
0
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1answer
33 views

How to use class equation for determining the center of $S_4$

How to use class equation for determining the center of $S_4$ $$|G|=|Z(G)|+\sum_x [G:C_G(x)]$$ So I guess I need to find $$|G|-\sum_x [G:C_G(x)]=|Z(G)|$$ Well $|S_4|=4!=24$ and $C_G(x)$ is the set ...
2
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1answer
28 views

transitive subgroups of the symmetric group on $2(d+1)$ elements: can I always do at least $d$ permutations?

I have a set of $2(d+1)$ elements which are labelled as pairs $\{e_i, a_i\}_{i=1}^{d+1}$, transforming under some transitive subgroup of the symmetric group. This can be thought of as a regular ...
3
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1answer
40 views

On the number of conjugacy classes in $S_n$.

Let $\sigma=(1,2,11)(3,4)(5,6,7,8,9)\in S_{12}$. I am willing to get the number of conjugates of $\sigma$. Clearly if $\tau$ be one such, then it must have the same cycle type. So in other words, we ...
3
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0answers
61 views

The Fifteen Puzzle and $S_n$ [duplicate]

I was studying permutation groups from the book "Abstract Algebra and Applications" by Karlheinz Spindler in which page 553 I came across the following interesting problem. It is on the famous "The ...
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1answer
35 views

Symmetric Polynomials and Automorphisms of Complex Polynomial Rings

I asked a version of this question earlier, but it was very imprecise and poorly formatted, so I decided to create a new question. Suppose we have an ordered set of $n(n-1)/2$ distinct polynomials ...
1
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1answer
60 views

Product of $n(n-1)/2$ polynomials of the same degree is symmetric

I am trying to prove a simple fact about polynomials in the multivariate polynomial ring $\mathbb{C}[x_1,x_2,...x_n]$, for $n \gt 3$ but I've been getting stuck. EDIT: After a comment by Tad I ...
1
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1answer
48 views

$SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong \mathbb A_4$

I am trying to prove that the quotient group $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3))$ is isomorphic to $\mathbb A_4$. I could show that $SL_2(\mathbb Z_3)$ has $24$ elements (one can see this ...
1
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1answer
43 views

Composition of groups

Let's say we have a system of interacting particles that can divided into two populations. The symmetry group of each population is $G$, and the two populations are identical, so that I can exchange ...
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2answers
86 views

How to prove that a given group is isomorphic to Sym(4)?

Given a specific group with 24 elements, I want to prove that it is isomorphic to Sym(4). To begin with, I calculate the orders of my group's elements and they come out as in the order statistics for ...
0
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1answer
17 views

Need help understanding the precise meaning of “unique factorisation of disjoint cycles”

Below is taken from my linear algebra course lecture notes: Some facts about permutations of $\{1,2,\dots,n\}$: Every permutation is a product of disjoint cycles which commute. For example ...
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0answers
19 views

Cycle Structure of a Permutation Based on the Binary Representation

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, ...
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2answers
49 views

Representing a 5-cycle as a product of transpositions

Dr. Pinter's "A Book of Abstract Algebra" shows that: $$(12345)$$ can be written as the following product of transpositions: $$(54)(53)(52)(51)$$ How can the first representation, $(12345)$, be ...
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0answers
27 views

Permutation module $M^\lambda$ as induced module

If we let $r$ be a natural number, $\lambda$ be a partition of $r$, $\Sigma_r$ be the symmetric group on $r$ numbers, we can define the following $K\left[ \Sigma_r \right]$-module: $M^\lambda := ...
3
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0answers
33 views

Conjugation of permutations

In the group $S_n$ I usually use the fact that if $(a_1 a_2 \dots a_r) \in S_n$ is an r-cycle and $\sigma \in S_n$ then $\sigma (a_1 a_2 \dots a_r)\sigma^{-1} = (\sigma(a_1)\sigma(a_2) \dots ...
2
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0answers
32 views

Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
1
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1answer
22 views

Prove this result relating to the sign of a permutation

Suppose that $\phi \in S_n$ is a permutation. Suppose also that $\psi = \phi \circ (i,j),$ where $1 \leq i, j \leq n.$ Why does it follow that sign$(\phi) = $ $-$sign$(\psi)$?
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0answers
38 views

Equivalence of irreducible representations of special linear group SL$(n)$ via those of GL$(n)$ and invariant total anti-symmetric tensor

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. Theorem 13.14 discusses the the equivalence of irreducible representations of special linear group ...
4
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1answer
73 views

Group acting on its set of subgroups by conjugation

I'm pretty sure for the first $H$, the Stabiliser is all of $S_4$ due to the normality of $V_4$, and so the Orbit is just $V_4$. For the second $H$, I have that the Stabiliser is $H$, as $4$ has to ...
2
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0answers
52 views

Idemptent of Young Tableaux

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. I understood that different Young Diagrams corresponds to inequivalent irreducible representations of ...
0
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0answers
9 views

Modular arithmetic of rotation a triange

A triangle is an n-gon with 3 rotational symmetries and 3 reflection symmetries. Each rotation is 120 degrees. Suppose the triangle begins with initial angle 240 degrees and rotates through 240 ...
4
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1answer
90 views

Find the smallest $n$ such that $\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$ is isomorphic to a subgroup of $S_n$

Let us consider the group $A=\mathbb Z_2\times \mathbb Z_2\times \mathbb Z_2$. Find the smallest positive integer $n$ such that $A$ is isomorphic to a subgroup of $S_n$. My thought. Since ...
2
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0answers
35 views

Trade-off among symmetries

Take a set $X \in \mathbb{R}^2$ of nonzero measure $\mu(X) \neq 0$. I am attempting to design a set that has the following symmetries (continuous or discrete) $1.$ Scale symmetry $2.$ Rotation ...
1
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3answers
51 views

Group of $2n$ elements, $n$ odd, is not simple

Problem Let $n \geq 3$ be an odd number and let $G=\{1,...,2n\}$ be a group of order $2n$. Let $\phi:G \to S_{2n}$ be the morphism defined by $\phi(g_i)(g_j)=g_ig_j$ and let $H=\phi^{-1}(A_{2n})$. ...
0
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0answers
33 views

How do I prove that the symmetric group $S_p$ where p is prime can be generated by any transposition and any p-cycle?

I am at a complete loss as to how to even begin. I think it has something to do with the fact that any p-cycle can be represented by $(123...p)$?
0
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0answers
17 views

Why is the k-th convolution of $P_S$ is equal to ${P_S^k}$

For a random walk using transpositions on $S_n$, how can it be explained that the k-th convolution of $P_S$ is equal to ${P_S^k}$. They look to be the same intuitively but how can it written ...
4
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1answer
67 views

Quotient space of $\Bbb C^n$ obtained by action of $S_n$

Consider the action of $S_n$ on $\mathbb{C^n}$ given by: $$\sigma(x_1, x_2, \cdots,x_n) = (x_{\sigma(1)}, x_{\sigma(2)}, \cdots,x_{\sigma(n)}).$$ What is the quotient space of $\mathbb{C^n}$ obtained ...
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0answers
23 views

Check the order of a subgroup of the alternating group $A_n$

Question : For a positive integer $n \geq 4$ and a prime $ p \leq n$. Let $U_{p,n}$ denote the union of all p- syllow subgroups of the alternating Group $A_n$ on n letters . Also let $K_{p,n}$ denote ...
2
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0answers
21 views

why is the sum over even and odd permutations the same?

let $m$ be an $n \times n$ matrix (over $\mathbb{R}$,say) and for a permutation $\sigma \in S_n$ define the monomial: $$ P_\sigma(M) = \prod_{j=1}^n m_{j,\sigma(j)} $$ let $\tau$ be an odd ...
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1answer
27 views

Finding the two different conjugacy classes in $A_n$ after splitting criterion

Suppose we have a group $A_n$ for some $n$ (maybe take $A_5$ as an example). We find the conjugacy classes of $S_n$ which are determined by cycle type. Then we use the splitting criterion ...