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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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number of permutation of S4 as product of two disjoint cycles each of length 2

There was a problem of finding out the number of permutations of order 2 in S4. There are two cases. case-1 permutation of single cycle of length 2. case-2 permutations of two disjoint cycles ...
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55 views

Does $\langle ( 1 ,3), (1, 2 … ,10)\rangle$ generate the group $S_{10}?$ [duplicate]

Does $\langle ( 1, 3), (1 ,2 ..., 10)\rangle $ generate the group $S_{10}$ ? I think that's it doesn't because every use of $(1, 3)$ makes a "jump" between at least two numbers. So we can get for ...
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37 views

Why does group action by conjugation on Sylow subgroups define a representation

I am studying representations and I am stumbled upon this: Take the dicyclic group of order $12: G=\Bbb Z/3\Bbb Z \rtimes \Bbb Z/4\Bbb Z$. The action of G on its 2-Sylow subgroups appearantly gives ...
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Question on irreducible character of regular representation of the symmetric group

Consider the symmetric group $S_n$ acting on $A=\{1,..,n\}$, for any nonnegative integer $k\leq n/2$, denote $A_k$ to be the collection of all $k$-element subsets of $A$. Let $\chi_k$ be the character ...
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29 views

The coset corresponding to permutation $(123)$ in $\Bbb Z /3\Bbb Z$.

We know that $V_4 \triangleleft A_4$ so $ A_4/V_4 \cong \Bbb Z /3\Bbb Z$. The coset corresponding to permutation $(123)$ is $(123)V_4$. Is it corresponding to $\overline{1}$ or $\overline{2}$ in $\...
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29 views

Using wreath products to find stabilisers of a partition of a set

I have the following example, that uses the wreath product to find the stabilisers of a partition. I don't understand how the wreath product does this though. I can recite the definition of a wreath ...
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1answer
45 views

The size of a conjugacy class in the symmetric group

I'm trying to prove the formula for the size of a conjugacy class in $S_n$. here is my try, Suppose the permutation whose conjugacy class size is to be found has a number of $k_i$ cycles of lengths $...
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1answer
35 views

how to check if there is an automorphism mapping between two conjugacy class

Let $G\le S_n$ be a permutation group and suppose that $C_1,C_2$ are two distinct conjugacy classes that have the same cardinality and is represented by a permutation of the same cycle-type. My ...
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1answer
27 views

Show that $S_n \cong A_n \rtimes C_2$ [duplicate]

I want to show that $S_n \cong A_n \rtimes C_2$. Take a transposition $\tau \notin A_n$. Then it is clear that $$\langle \tau\rangle \cap A_n = 1$$ $$A_n \tau = S_n$$ $$A_n \unlhd S_n$$ and thus ...
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Is knowing the character enough to determine the representation

Let V be a complex représentation of finite dimension of the symmetric group $S_n$ and $χ$ the character of $V$. prove $V=V\otimes_{\Bbb C}\varepsilon$ iff $χ(\sigma)=0$ for every odd permutation, ...
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Sum of signature of elements of $S_n$ is $0$

I saw in a proof that for each $n > 1$, the symmetric group $S_n$ satisfies $$\sum_{g\in S_n} \varepsilon(g) =0,$$ where $\varepsilon$ is the signature. Is that true? I checked it is true ...
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1answer
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The difference between permutation representation and standard representation of symmetric group $S_3$

I was reading this paragraph from Fulton & Harris : But I could not understand the difference between permutation representation and standard representation and how they are complement and why ...
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1answer
76 views

Is there a good way to show that the order of element in $S_7$ are at most $12$?

The only solution would be going through all cycle types of all permutations which is a lot of work. Is there any smarter solution than this one? Thank you in advance!
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Decomposition of tensor product of permutation representation of $S_n$.

Please be kind to me - I'm a combinatorist so this question might be a bit naive... If $U$ is the representation space of the permutation representation of $S_n$, is there any known decomposition ...
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29 views

Prove that a permutation is a $p$-cycle

I have difficulties proving the following theorem: Let $σ ∈ S_n$ satisfy $σ \neq (1)$, and $σ^p = (1)$, where p is a prime number such that $\frac{n}{2} < p ≤ n$. Prove that σ is a p-cycle. ...
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What physical features of the tetrahedron prevent its symmetry group from being all of $S_4$?

This is an exercise from the book "Visual Group Theory" by Nathan Carter. Exercise $5.26$. As you know from the chapter, the symmetry group for the tetrahedron is $A_4$. We can think of it, as you ...
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22 views

Partitioning evens as sum of evens

Take the set $\{a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8\}$. We can partition according to rules. Every member in the partition has even number of elements. Every member in partition have to be consecutive. ...
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24 views

The proof of Frobenius Formula in Fulton and Harris “representation theory a first course” .

I was reading the proof of Frobenius Formula in Fulton and Harris "representation theory a first course", but it was hard to me to follow the book, can anyone recommend a better source for me that ...
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53 views

Show that for $n\ge 2$, $S_n$ can not be imbedded in to $A_{n+1}$ [duplicate]

I asked a question about this yesterday (Show that $S_n$ is never isomorphic to a subgroup of $A_{n+1}$ for $n\ge 2$) and it was marked as duplicate, but the links given did not provide a clear ...
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Show that $S_n$ is never isomorphic to a subgroup of $A_{n+1}$ for $n\ge 2$

in my notations, $S_n$ is the symmetric group and $A_{n+1}$ the alternating group. I showed that if $n$ is even it is never possible because you need to have $n! | n! \cdot\dfrac{n+1}{2}$. So we ...
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Symmetric group, give permutation such that $\tau \sigma \tau^{-1} = \sigma^5$

I have the permutation $\sigma = (123)(58)$ in the symmetric group $S_8$ of degree $8$. I wish to find a permutation $\tau$ being an element of $S_8$ such that: $\tau \sigma \tau^{-1} = \sigma^5$ I'...
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33 views

Number of conjugates in symmetric group

I have the permutation $\sigma=(123)$ in the symmetrical group $S_5$ of degree $5$ and I want to find the number of conjugates to sigma in $S_5$. I know that the conjugates to sigma will be the ...
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1answer
67 views

Using Pólya counting to find number of conjugacy classes of $S_3$

So I know that $S_3$ has three conjugacy classes. However, I was reading about Pólya counting today, and am wondering how Pólya counting could be used to derive the number of conjugacy classes for $...
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1answer
48 views

A question on an identity involving partition

Let $n$ be a natural number. Let $\lambda \mapsto n$ , be a partition of n. So $\lambda=(\lambda_1, \lambda_2, \ldots ,\lambda_k)$, with $\lambda_1\leq \ldots \leq \lambda_k$, and $\sum_{i=1}^k \...
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1answer
36 views

Subgroup for Symmetric Groups

Let $A=\{2,4,5\}$ and let $H=\{F\in S_q \mid F(A)=A\}$ (a) Show that $H$ is a subgroup of $(S_q,\circ)$ (b) What is the order of $H$, $|H|=?$ Not quite finding out how to do the first ...
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1answer
34 views

Is it possible to “mod” the action of a symmetric group on a symmetric operad?

I am relatively new to category theory, so only have a rough understanding of the technicalities behind operads. My understanding is that symmetric operads are defined so that they are "nicely" acted ...
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74 views

Landau’s Function

Show that for all L(n)<2^n for all n ∈ N Where Landau’s function L(n) is defined for every n ∈ N to be the largest order of an element of Sn. I have proven by induction, that n<2^n for all n ∈...
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Eigenspaces of the Cayley Graph Cay$(S_n,T_n)$ on adjacent transpositions

Consider the Cayley graph G = Cay$(S_n,T_n)$ where $S_n$ is the symmetric group and $T_n = \{(i,i+1) | 1 \leq i \leq n-1\}$ is the set of adjacent transpositions. G is sometimes called the ...
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1answer
52 views

How many elements of order $4$ in $S_{7}$?

I'm aware that there a questions like this but I would like to check whether my approach is good. I know the order of disjoint cycles is the product of their lengths. Thus, to get order $4$ with $S_{...
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44 views

Prove if the the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not are isomorphic

I'm trying to check if the groups $A_4$ and $S_3 \times \Bbb Z_2$ are or not isomorphic. How can I check if they are? I'm trying to understand how can I generally prove an isomorphism with this kind ...
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53 views

$G$ non-centerless group. How is $Z(\operatorname{Aut}(G))$ made?

Let $G$ be a (possibly infinite) non-centerless group, i.e. such that $Z(G) \ne \lbrace e \rbrace$. Left and right multiplications establish the subgroups $\Theta:=\lbrace \theta_a \mid a \in G \...
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Wassertein and symmetry

So here's a scenario: I have points $(\mu_1^j,\mu_2^j)$ and I associated them the following distribution $$\rho_j=1/2\delta_{\mu_1^j}+1/2\delta_{\mu_2^j}$$ These have symmetry (exchanging $\mu_1^j$ ...
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1answer
44 views

Certain Isomorphic Representations of the dihedral group $D_{3}$

Using the following presentation of the dihedral group $D_{3}$ \begin{equation} D_{3} = \left\langle r,s \mid r^{2} = s^{2} = (rs)^{3} = e \right\rangle \end{equation} There is one (...
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Intersections of row stabilizers of λ-tabloids

Let λ be a partition of n. For which case there exists a pair of Young λ-tabloids such that the intersection of their row stabilizers is trivial?I know the condition implies that the conjugate of λ ...
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1answer
23 views

constructing new genereating set for $S_n$ [closed]

Is it true that $S_n$ is generated by the transposition $(12)$ and the set of $3$-cycles $\{(123),(124),\dots ,(12n)\}$ according to the splitting lemma of $Z_2$ and $A_n$?
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1answer
36 views

Complex group algebra of $S_3$ isomorphic to $\mathbb{C}[x]/(p)$?

For the symmetric group on two letters, $S_2$, there exists an isomorphism from the complex group algebra $\mathbb{C}[S_2]$ to the complex polynomial algebra $\mathbb{C}[x]/(x^2 - 2x)$ by taking $e + (...
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51 views

Let $H=\{\alpha \in S_7 :\alpha(3)=3\}$ and $K=\{\alpha \in S_7:\alpha(5)=5\}$. Prove that $H\cong K$.

Question: Let $H=\{\alpha \in S_7 :\alpha(3)=3\}$ and $K=\{\alpha \in S_7:\alpha(5)=5\}$. Prove that $H\cong K$ I'm not really able to do much here, I know I have to find some isomorphism but I'm ...
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2answers
40 views

Find number of flips of transposition $(i \ j)$.

Suppose that $(i \ j) \in S_n$ is a transposition and that $i < j$. How would it be possible to find an expression for the number of flips of $(i \ j)$?
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32 views

How can I show that a representation of $S_n$ is reducible

Considering the representation of $S_n$ where the objects being permuted are the basis vectors of an $n$ dimensional vector space $$ |1 \rangle, |2 \rangle, \:... \:, |n \rangle$$ If the ...
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Group of Proper Symmetries of Painted Cube

What is the proper symmetry group of a cube in which three faces, coming together at one vertex, are painted green and the other faces are red? I know that the axis of rotation for which the ...
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1answer
93 views

Show that S6 has at least 60 subgroups of order 4

Show that $S_6$ has at least $60$ subgroups of order $4$. [Hint: Consider cyclic subgroups generated by a 4-cycle (such as $\langle(1234)\rangle$) or by the product of a 4-cycle and a disjoint ...
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1answer
36 views

What is the span of the symmetric group $S_{5}$?

My professor while writing the character table of $A_{5}$, uses that the span of $S_{5}$ is $\{e_{1} - e_{2}, e_{2} - e_{3}, e_{3} - e_{4}, e_{4} - e_{5} \}$ but I do not know why this is the span of $...
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2answers
35 views

If $\alpha$ and $\beta$ are disjoint, so are $\alpha$ and $\beta^k$?

If $\alpha$ and $\beta$ are disjoint permutations, so are $\alpha$ and $\beta^k$. I'm working on the group of permutations of a finite set $X$ with composition. The power is defined as usual using ...
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2answers
34 views

Identity from repeated function composition

Do functions exist such that $f^n(x)=x$ for values $n > 2$? For $n=2$ we have $-x$ and $1/x$, and for $n=3$ we can show $1/(1-x)$ is a solution. I assume that $n$ must be prime and preferably ...
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1answer
53 views

Writing $2$-cycles as a product of adjacent 2-cycles.

Problem: Consider the arbitrary 2-cycle $(a\ b)$ from $S_n$. Find a way to write this permutation as a product of adjacent 2-cycles. What I do know: A transposition is a single cycle of length 2. An ...
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1answer
38 views

If $n\geq3$ and $G\leq S_n$ is 2-transitive, then $G\cap A_n$ is transitive.

A subgroup $G\leq S_n$ of the symmetric group is said to be 2-transitive if it acts transitively on the set of ordered pairs of distinct elements of $\{1,\ldots,n\}$. It is said to be 2-homogeneous if ...
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1answer
109 views

Unexpected proof that $\alpha!$ divides $k!$ if $\alpha_1 + \dots + \alpha_n = k$.

Let $\alpha = (\alpha_1,\dots, \alpha_n) \in \mathbb{N}_0^n$ be a multiindex with $\alpha_1 + \dots + \alpha_n = k$. Let $\alpha! = \alpha_1! \dots \alpha_n!$ with the convention that $0! = 1$. I ...
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82 views

Visual representations of groups (in their symmetric groups)_part#2

Background In this post, I have shown that a plausible visual representation of a group $K$ in $\operatorname{Sym}(K)$ can be established, where $\operatorname{Aut}(K) \setminus \lbrace \iota_{\...
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1answer
69 views

why splitting lemma fails for nonabelian groups?

Like the title said, I want to ask why does it fail. The claim and the counterexample here https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Splitting_lemma.html are very ...
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0answers
38 views

$A_n\leq G\leq S_n$ implies $G=S_n$ or $G=A_n$?

Is it true that if $A_n\leq G\leq S_n$ implies $G=A_n$ or $S_n$? Here is my argument: Since $|G|=m$ divides $n!$ we have $mk=n!$ for some $k$. But $|A_n|=n!/2\leq n!/k$, hence $k\leq 2$, hence $m=n!/k$...