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Questions tagged [symmetric-groups]

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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Relation between characters corresponding to hook partitions

Let $(n-k,1^k)$ be a hook partition. I want to know if there is a relation involving any two, or maybe all three, of the quantities: $$\chi_{(n-k,1^k)},\quad \chi_{(n-k-1,1^k)},\quad \chi_{(n-k,1^{k-...
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1answer
24 views

Trying to understand a notation $S_{\lambda} w S_{\mu}$.

I am reading the paper (arXiv:1605.08545v5). There is a notation $S_{\lambda} w S_{\mu}$ on page 61 before the formula (27). It is said that $w$ is of maximal length in $S_{\lambda} w S_{\mu}$. Here $\...
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1answer
70 views

$S_7$ cannot be written as a union of fewer than $459$ cyclic subgroups.

Prove that $S_7$ cannot be written as a union of fewer than $459$ cyclic subgroups. I have no idea about this. I know that $S_7$ has $7!$ elements. However, this has not been helpful. We have not ...
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1answer
27 views

Number of conjugacy classes in permutation group $ S_n $ for some n.

I was asked to find out the number of conjugacy classes in the permutation group $ S_6 $.It is 11.But for large n it is difficult to find out .so I want to know is there any way to find out the ...
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2answers
39 views

Determine the class equation of the tetrahedral group

This is a question from Artin's algebra textbook. The tetrahedral group of rotations has 1 element of order 1, 8 elements of order 3 (rotations of $120^°$ around a vertex), and 3 elements of order 2 ...
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35 views

Number of conjugates of $(12)(34)(56)(789) \in S_{10}$

Number of conjugates of $(12)(34)(56)(789) \in S_{10}$ This is how I calculated it and got $840$ as a result: $${10\choose 2}\cdot{\frac{2!}{2\cdot3!}}\cdot {8\choose 3}\cdot\frac{3!}{3}$$ What I ...
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1answer
32 views

Is there a name for the quotient of the symmetric group by the finitary symmetric group?

The finitary symmetric group on a set $S$ is the group of permutations that only move a finite set points. That is: $$FSym(S) = \{\phi:\{s : s \in S, \phi(s) \neq s\}\text{ is finite}\}$$ This is a ...
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63 views

How similar are permutation groups that are isomorphic as abstract groups?

Let's say that two permutation groups $P_1$ and $P_2$ are isomorphic as abstract groups, but not necessarily permutation isomorphic. How similar will $P_1$ and $P_2$ be, and how much structure will ...
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35 views

For $n>1$, let $H$ be the set of all products in $S_n$ of a multiple of four transpositions. Show $H=A_n$.

I'm reading "Contemporary Abstract Algebra (Eighth Edition)," by Gallian. This is Exercise 5.80 ibid. Answers that use only tools available in the textbook so far are preferred. For $n>1$, let $...
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1answer
45 views

Meaning of permutation representation

I'm following the discussion in Artin's Algebra. Artin defines a permutation representation of a group $G$ as a homomorphism from the group to the symmetric group:$$\varphi:G\rightarrow S_n$$ Then ...
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1answer
21 views

f interchanges $\alpha$ and $\beta$ but fixes $\gamma$ and $\delta$

$f=(\alpha \beta), \; g=(\beta \gamma), \; h=(\gamma \delta)$ Is there a way to solve this that avoids finding what the permutations in options are ?
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1answer
34 views

For which $n\in \Bbb N$ is $H_n:=\{\alpha^2\mid \alpha\in S_n\}\cong A_n?$

I'm reading "Contemporary Abstract Algebra," by Gallian. This is inspired by Exercise 5.73 and Exercise 5.74 ibid. I have a preference for answers using only the tools available in the textbook so ...
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2answers
48 views

Order of $\alpha=(124)(5439)(328)(1378542)\in S_{10}$

I know that the order of an element in a symmetric group is given by $\text{lcm}(a,b)$ where a and b are the length of the cycle. But not sure what to do with more then 2 cycles.
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19 views

Show $\nexists x\in S_7$ with $x^2=(1234)$ but there is at least two $x\in S_7$ with $x^3=(1234)$. [duplicate]

I'm reading "Contemporary Abstract Algebra," by Gallian. This is Exercise 5.48 ibid. and I want to answer the question using the tools available in the textbook so far. (A free copy of the book is ...
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0answers
26 views

The decomposition into two irreducible SU(4) representations

I read in the book "String Theory and $M$-theory: A Modern Introduction" that the eight components can be decomposed into two irreducible $SU(4)$ representations: $8=4 \otimes \bar{4}$. Is it correct ...
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260 views

Visual representations of groups (in their symmetric groups)

Given a group $G$, left and right multiplications establish the subgroups $\Theta:=\lbrace \theta_a \mid a \in G \rbrace \le \operatorname{Sym}(G)$ and $\Gamma:=\lbrace \gamma_a \mid a\in G \rbrace \...
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79 views

Symmetric group action on polynomial ring

Let the symmetric group $S_4$ act on $\mathbb R[x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4]$ by permuting the 1st $4$ variables and again permuting the last $4$ variables. We can restrict the action to the ...
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0answers
23 views

For given permutation, find solutions for $x^3=\sigma\in S_{9}$

We have the given permutation $$\sigma =\left({\begin{array}{*{20}c}1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 4 & 5 & 6 & 7 & 8 & 9 & 1 & 2 & ...
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35 views

Stabilizer in the definition of a Gelfand pair

I am following the textbook Representation Theory of the Symmetric Groups, by Tullio C.-S., Fabio S., and Filippo T., and am confused at the definition of a Gelfand pair. The definition is: Let $G$ ...
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24 views

How to compute quickly the order of the automorphism group of $A_4$

I'm recently got interested by a series of very cool videos by Math Doctor Bob, specifically the ones about to compute automorphism groups. The one puzzling me a bit is this one: Automorphism of A4 ...
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2answers
48 views

A difficulty in understanding an example in Vinberg.

The example is given below: But I have difficulties in understanding the following: 1- why $V_{0}$ is called $(n-1)-$dimensional subspace, I want a concrete example please? 2- Why if the ...
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1answer
98 views

Proving that $\text{Sym}(T)$ is a group without treating functions as sets

One of the points that Tim Gowers makes in this blog post, is that the set theoretic definition of functions achieves nothing, that it is possible to treat functions as fundamentally different objects ...
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1answer
31 views

Homomorphisms of $S_3$

What are all the groups (not counting the isomorphic ones) that can be a homomorphic image of $S_3$? So here's what I have come up with so far: First of all, $S_3$ has 3! = 6 elements, which are $\{ ...
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1answer
70 views

Equation with Galois group twisted $S_{3}$

I note that a Galois group is not just a Galois group. Let $r_{1}$, $r_{2}$, $r_{3}$, $r_{4}$, denote the roots of a quartic equation. Then $x^4-5x^2+6$ has Galois group $Z_{2}^2$, where the ...
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2answers
26 views

Verifying the calculation of the number of conjugate elements

I would like to verify my understanding: Consider action of $S_7$ on itself by conjugation. I'm trying to compute: $Stab_{S_7}((1 2))$ $Stab_{S_7}((1 2 3 4 5 6 7))$ $Stab_{S_7}((1 2 3)(4 5 6))$ I'm ...
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36 views

Proving that $|S_8 \cdot a|=\binom{8}{4}\frac{4!}{4}$

So I came across with the following exercise: How many elements are conjugated with $a=(3\ 5\ 1\ 6)$ in $S_8$? The solution was: we are looking for $|Stab(a)|=|\{b\in S_8 : bab^{-1} = a\}|$ ...
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2answers
72 views

How do I find the order of a bijection?

Question: List all bijections (permutations) from $\{1, 2, 3\}$ onto $\{1, 2, 3\}$. Find their order and sign. I understand there will be n! permutations, namely: $ \begin{Bmatrix} 1 & 2 &...
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2answers
80 views

Is $A_4 \times \mathbb{Z}_3$ isomorphic to $S_3 \times S_3$?

Ιs $A_4 \times \mathbb{Z}_3$ isomorphic to $S_3 \times S_3$? I am trying to find an element of $S_3 \times S_3$ which has an order, let's say $a$, and $A_4 \times \mathbb{Z}_3$ has no element of ...
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2answers
34 views

List of symmetric group elements from the usual presentation

Let $S_n$ be the symmetric group of $n$ letters, generated by elements $s_i$ for $ 1 \leq i \leq n -1$ with relations $s_is_{i+1}s_i = s_{i+1}s_is_{i+1} $ and $s_i,s_j$ commute if $|i-j| > 1$. I ...
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1answer
54 views

Let $\phi: S_n \to G$ be a homomorphism, where $|G|$ is odd. Prove that $\phi$ must be trivial.

The Problem: Let $n$ be a positive integer, let $G$ be a finite group with an odd number of elements, and let $\phi: S_n \to G$ be a homomorphism. Prove that $\phi$ must be trivial, i.e., $\phi(\...
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2answers
59 views

Group generated by $(1234567)$ and $(26)(34)$ is of order 168.

Let $G$ be a subgroup of $S_7$ generated by $(1234567)$ and $(26)(34)$. Show that $|G| = 168$. This is a question from Algebra by Hungerford (page.112, exercise 15). And I just have no idea of ...
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1answer
49 views

f irreducible polynomial with $p-2$ real roots $\Rightarrow$ $Gal(\mathbb{Q}_{f}/\mathbb{Q}) \cong S_{p}$

I have no idea what to do to show the following Let $p\ge 5$ be a prime number 1) Let $H\subset S_{p}$ be a subgroup of the symmetric group. Assume that $p$ divides the order of $H$ and that $H$ ...
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45 views

Unique Elements from Subsets of Symmetric Group

Good morning to you all! I have math question, plz help me out! We take a subset $A$ from a symmetric group $S_n$ (n elements). Now consider all elements of $B$ composed of $a * b$ where $*$ is the ...
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2answers
65 views

Determine whether the following elements of $S_6$ are of the form $\sigma^2$.

The Problem: Let $S_6$ be the symmetric group on six letters. Determine whether the following elements of $S_6$ are squares (i.e., of the form $\sigma^2$). (a) $(1 \hspace{1mm} 2 \hspace{1mm} 3 \...
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1answer
54 views

Are given groups isomorphic

Are there isomorphic groups between $\Bbb{A}_4\times \Bbb{Z}_3,\Bbb{D}_{18},\Bbb{D}_{9}\times \Bbb{Z}_2,\Bbb{S}_3\times \Bbb{S}_3$? Where $\Bbb{D}$ is the dihedral group,$\Bbb{A}$ alternating group ...
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1answer
34 views

What are the elements in the subgroup generated by $\langle(1234)\rangle$ in $S_4$? [closed]

What are the elements in the subgroup generated by $\langle(1234)\rangle$ in $S_4$?
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23 views

Solving doubleintegral $\int\int f(g(x,y)) dy dx$ over annulus $D$ in $R^2$ where $g$ is Isometry

I want to solve certain triple integral $\int\int f(g(x,y)) dy dx$ over annulus $D$ (concentric about origin) in $R^2$ where $g$ is an Isometry of a plane and $f$ is a given function. Because of the ...
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1answer
25 views

cardinal of conjugacy class of $(123)(456)$ in $A_7$ [duplicate]

So it is not hard to see that the conjugacy class of $(123)(456)$ in $S_7$ has cardinality $\frac{7!}{3\cdot 3\cdot2}=280$. But for instance to go from $(123)(456)$ to $(132)(456)$ you have to ...
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1answer
34 views

If $|G|<\infty$ acts transitively on a set $X$ with $|X|=10$ then $G$ has an element of order $5$

Let $G$ be a finite group that acts transitively on a set $X$ with $|X|=10$. Show that $\exists g\in G$ of order $5$ There is a homomorphism $\phi:G\mapsto S_{10}$ that sends $g\in G$ to its ...
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34 views

What are the most efficient date systems with temporal symmetry?

Most date or time systems have an Epoch, or privileged "starting" point. For example, the Epoch of the Gregorian calendar is 1 A.D. The Epoch of Unix time January 1, 1970. These Epochs introduce a ...
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3answers
69 views

Proving $(a b)(c d)=(a b c)(b c d)$ [closed]

I was reading a book in Group theory and they said that $(a\, b)(c\, d)=(a\, b\, c)(b\, c\, d),$ but they did not prove it, for some reason (maybe it's too obvious but i can't seem to see it). How ...
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1answer
42 views

What are the conjugacy classes in $A_5$?

I'm new to group-theory and to the Alternating groups. In my book I got asked the following questions: What are the conjugacy classes in $A_5$? Where should I start? What does it mean, "conjugacy"?...
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1answer
30 views

Proving $\{gY:g\in X\}=\{Yg:g\in X\}$ [duplicate]

I was trying the study a new subject in groups (from a book I have). I came acorss with the following question: Consider: $X=S_4$ $Y=\{e,(12)(34),(13)(24),(14)(23)\}$ How can I prove that: $$\{gY:...
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53 views

Prove that there exists a semi-orthogonal $U$ such that $U^TAU=B$, where $A$ and $B$ are positive-definite symmetric matrices.

Let there be a semi-orthogonal matrix $U \in \mathbb{R}^{m\times n}$ such that $U^TU=I_n$ if $m > n$ If $A \in \mathbb{R}^{m\times m}$ and $B \in \mathbb{R}^{n\times n}$ are positive-definite ...
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1answer
189 views

What is the number of squares in $S_n$?

Let $$X_n=\{\sigma\mid\sigma=\tau^2 \text{ for some }\tau\in S_n\}.$$ What is the cardinality of $X_n$? For example, permutation $(12)(3456)$ is not a square in S_n. I know that $X_n=A_n$ for $n\leq ...
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1answer
55 views

Is there exist a group homomorphism from the symmetric group $S_n$ to $S_{n-1}$ for $n \ge 5?$

Does there exist a group homomorphism from the symmetric group $S_n$ to $S_{n-1}$ for $n \ge 5?$ My attempt: I think not, because for $n \ge 5$ , $A_n$ is the only normal subgroup of $...
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1answer
28 views

About the Proof for “the Order of a Permutation $\sigma$ is the $lcm$ of the Orders of its Disjoint Cycles.”

About the lemma below: The order of a permutation $ \sigma \in S_n $ is the least common multiple of the orders of its disjoint cycles. I didn't get the proof which consists of the following ...
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1answer
69 views

Showing $S_{n}$ is not solvable for $n\geq 5$

I'm going through a physicist viewpoint of Galois theory and a found a lecture by Takeuchi here. In the slide 47 he showed that $S_{n}$ is not solvable for $n\geq 5$. This is the proof Let $G$ be a ...
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1answer
55 views

The centralizer of a cycle is the group generated by the elements of $S_n$ that are disjoint from the cycle and the group generated by the cycle.

I am having a hard time trying to prove that the centralizer of a cycle is the group generated by the elements of $S_n$ that are disjoint from the cycle and the group generated by the cycle. It is ...
1
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1answer
27 views

Representatives for all conjugacy classes of elements of order 15 in A11

I am trying to find representatives for all conjugacy classes of elements of order 15 in $A_{11}$. It is not hard to see that $(12345)(678)$ and $(12345)(678)(9 10 11)$ are the representatives for ...