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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Groups isomorphic to $S_{4}/N$

Let $G = S_4$ be a group, $N = \{1, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)\}$ a normal subgroup of G. It's easy to see that $G/N$, the set of cosets is $G/N = \{a, b, c\}$, where $$a = \{(1), (1, ...
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1answer
11 views

Finding all permutations which satisfy given condition

In a symmetric group $S_n$ find number of permutations $P$ such that in the disjoint cycle decomposition of $P$ , length of cycle containing $1$ is $k$ . Here's my attempt at this . I found number of ...
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0answers
18 views

On the probability distribution of iterated permutations

I have this little problem that has been nagging me for a couple of months now. It occurred to me when considering the fairness of card shuffling methods. Here's my best attempt at formalizing it: ...
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0answers
17 views

Conjugate classes of Symmetric group, $S_n$ and partition number of $n$.

In the book "Topics in Algebra", 2nd edition, By I.N. Herstein, the following lemma is given on page 89, Lemma 2.11.3 :The number of conjugate classes in $S_n$(the symmetric group of order $n$) is ...
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26 views

What is the probaility that two random permutations have same order?

I am interested in the orders of random permutations. Since the law of the log of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one ...
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1answer
47 views

Prove that (01) and (01…n-1) generate Sn? [duplicate]

Show that every element of Sn can be written as an arbitrary product of the elements (01) and (01...n-1). I understand that this can be solved using induction, and I've set up my base cases. ...
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1answer
15 views

Why is the Young symmetrizer non-zero?

Suppose $\lambda$ is a partition of the natural number $n$ and $T$ is a standard Young Tableaux of shape $\lambda$. Let $$P_{\lambda}:=\lbrace g\in S_n:g\text{ preserves the rows of }T\rbrace$$ and ...
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$S_X = \{f(x) = x : \text{ bijective} \}$. prove $S_x$ is isomorphic to $S_n$

Aright, to start $S_n$ is the Symmetric group and $S_X = \{x_1, x_2, \ldots x_n\}$. Going through the mapping $\phi(S_X) \to S_n$, I'm not sure how I'd show this mapping and the first thought that ...
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56 views
+50

Easy way to prove a relation in $S_n$

I need to prove that, denoting $s_i = (i, i+1) \in S_n$ (transposition), and for any $w \in S_n$, we have that if $l(s_i w s_j)=l(w)$ and $l(s_i w) = l(w s_j)$, then $s_i w= w s_j$. I know a proof ...
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13 views

Computing character for the partition $(2, 2, 1, 1)$ using Murnaghan–Nakayama rule

I am trying to understand an example from Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. The group ...
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18 views

Understanding the Murnaghan–Nakayama rule

I am trying to understand the Murnaghan–Nakayama rule as it is described in Normal Subgroup Reconstruction and Quantum Computation Using Group Representations by Hallgren et al. Here is the ...
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1answer
36 views

Permutation representations of symmetric group of small order

Thanks for any comments or help. What is the list of faithful permutation representations of $S_k$ of degree at most $n=2k$ for $k=3,4,5,6$? Is it possible to find its centralizer in each case?
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10 views

Maximal set of disjoint prime cycle permutations on $n$ elements to generate $\prod^t_i S_{N_i} \wr D_{m_i} $

How can I determine the maximal set of disjoint prime cycle permutations on n elements to generate $\prod^t_i S_{N_i} \wr D_{m_i} $? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the ...
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17 views

Automorphism group of planar graphs

I am going through Constructive Approach to Automorphism Groups of Planar Graphs by Klavík et al. It has shown that the automorphism group of a planar graph $G$ is as follows. $$ \text{Aut} ...
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1answer
19 views

Are subgroups of automorphism groups of trees direct product of symmetric, cyclic and dihedral groups?

My question is triggered by my confusion with the notation $\Psi$ in Constructive Approach to Automorphism Groups of Planar Graphs by Klavík et al. The notation $\Psi$ was first used expressing ...
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2answers
34 views

Abstract finite group vs. finite group

What is the difference between the definitions of abstract finite group and finite group? I have some exposure to the finite group theory. But I came to know about abstract finite group from ...
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0answers
13 views

Interpreting the table of classification of the partitions of $n$

I am going through A NON-RECURSIVE EXPRESSION FOR THE NUMBER OF IRREDUCIBLE REPRESENTATIONS OF THE SYMMETRIC GROUP $S_n$ by AMUNATEGUI. In table I, the classification of the partitions of n according ...
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0answers
8 views

Relation between $\prod^t_i S_{N_i} \wr D_{m_i} $ and $A_{\sum^t_i N_i m_i}$?

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a subgroup of the symmetric group $S_{\sum^t_i N_i m_i}$. Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ ...
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0answers
13 views

Wreath product of subgroup with symmetric group

I am going through Generic Quantum Fourier Transforms by Moore et al. I would like to put the screenshot of the section I am confused about below. So, Why does $H$ need to be of size $poly(n)$ for ...
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1answer
30 views

Term for 'distance' of two elements in a permutation

Consider the group $G := \langle (1,3), (1,2,3,4) \rangle$. This group is of order $8$ and its elements are $$G := \{ (), (1234), (24), (12)(34), (13), (14)(23), (13)(24), (1432) \}.$$ Each of ...
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8 views

Can there be a tower like $1 = A_0 \triangleleft \ldots \prod^t_i S_{N_i} \wr D_{m_i} \ldots \triangleleft = S_{\sum^t_i N_i m_i}$?

$\prod^t_i S_{N_i} \wr D_{m_i}$ is a subgroup of $S_{\sum^t_i N_i m_i}$. Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral ...
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0answers
30 views

Easy to generate subgroups of the symmetric group $S_n$

Which subgroups of the symmetric group $S_n$ can be generated in polynomial or subexponential time?
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14 views

Is $\prod^t_i S_{N_i} \wr D_{m_i}$ a maximal subgroup of $S_{\sum^t_i N_i m_i}$?

How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ is a maximal subgroup of $S_{\sum^t_i N_i m_i}$? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ ...
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0answers
31 views

The size of the automorphism group of a graph

I am going through QUANTUM MECHANICAL ALGORITHMS FOR THE NONABELIAN HIDDEN SUBGROUP PROBLEM by Grigni et a. It is said on page 14 that the size of the automorphism group of a graph is either $1$ or ...
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0answers
6 views

Presentation of $\prod^t_i S_{N_i} \wr D_{m_i}$

I would like to determine the presentation of $\prod^t_i S_{N_i} \wr D_{m_i}$. My effort: The presentation of the symmetric group $S_{N}$ is as follows. $\langle s_1, \ldots, s_{N-1} | (s_i ...
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12 views

Minimal generating set of A5 [duplicate]

I need to show that $\langle(12)(34),(12345)\rangle = A_5$ the alternating group such that $n=5$ I've been trying to think of a way to show that this is a normal subgroup, hence as $A_5$ is simple ...
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16 views

Is $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal subgroup of $S_{\sum^t_i N_i m_i}$?

How can I determine whether $\prod^t_i S_{N_i} \wr D_{m_i}$ a normal subgroup of $S_{\sum^t_i N_i m_i}$? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ ...
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0answers
12 views

How to determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$?

How can I determine the order of $\sum^t_i S_{N_i} \wr D_{m_i}$ ? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of ...
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10 views

How to prove $\sum^t_i S_{N_i} \wr D_{m_i}$ non-Abelian?

How do I prove that $\sum^t_i S_{N_i} \wr D_{m_i}$ is a non-Abelian group? Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral ...
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2answers
64 views

Prove that $S_4$ cannot be generated by $(1 3),(1234)$

Prove that $S_4$ cannot be generated by $(1 3),(1234)$ I have checked some combinations between $(13),(1234)$ and found out that those combinations cannot generated 3-cycles. Updated idea: Let ...
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1answer
24 views

Suppose that $X$ is a nonempty subset of a set $Y$. Show that $S_X$ is isomorphic to a subgroup of $S_Y$

Suppose that $X$ is a nonempty subset of a set $Y$. Show that $S_X$ is isomorphic to a subgroup of $S_Y$. Updated idea: Define $f:S_X\rightarrow f(S_X)$ by $$f(\sigma x)=\sigma x,\forall x\in X$$ ...
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1answer
31 views

Find all permutations such that $\sigma=a\tau a^{-1}$

For (b) and (c) we note that $\sigma$ and $\tau$ have different parity so there cannot be any $a\in S_4$ that will fix that parity mismatch. For (a) we have the cycle $a^{-1}=(3 2 4)$ and it is ...
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43 views

Center of the group algebra of the symmetric group

How to prove that the center of the group algebra of the symmetric group is generated by 1-cycle conjugacy classes? I mean, that the center (consisting on class functions) is multiplicatively ...
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24 views

Is $C_{S_n}(g) = H \times S_m$?

Let $g$ be an element of the symmetric group $S_n$. If $c$ commutes with $g$, then $c$ permutes the set of $g$-fixed numbers in $\{1,\ldots,n\}$. Write $C_{S_n}(g)$ for the centralizer of $g$ and ...
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1answer
17 views

The product of two transpositions is a commutator

Let $n \ge 4$ and $u,v \in S_n$. Prove that $uv$ is a commutator. In other words, prove that there are $\alpha, \beta \in S_n$ such that $uv = \alpha \beta \alpha^{-1} \beta^{-1}$. This is ...
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1answer
27 views

Show that every $\sigma \in S_n$ is of the form $\sigma = \prod_i (1 \; \; x_i)$

Let $n \in \Bbb N$ and let $S_n$ denote the group of permutations of $\{1,2,...,n\}$. Prove that for all $\sigma \in S_n$, we have: $$\sigma = \prod_{i=1}^m (1 \ \ x_i), \text{ for some ...
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1answer
32 views

the number of the conjugacy class sizes of $S_n$

If $a$ has the cycle type $(k_1,k_2,...,k_t)$ and $b$ has the type $(l_1,l_2,...,l_t)$ which is different with the cycle type of $a$, then can the equality $|C_{S_n}(a)|=|C_{S_n}(b)|$ occur? I hope ...
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25 views

Discussion of $Z(A_4) = \{e\}$

I tried to answer the following question: Why does the fact that the orders of the elements of $A_4$ are $1,2$ and $3$ imply that $|Z(A_4)|=1$? My answer: Two cycles commute if and only if ...
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1answer
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How to show that there are $1121505$ conjugacy classes in $S_{61}$? [closed]

How to show that there are $1121505$ conjugacy classes in $S_{61}$? Can someone help me with this question. I have no idea how to count them. Second Question: When is a conjugacy class a ...
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1answer
22 views

The signature map is a homomorphism

From the following definition of the signature of permutations on the symmetric group on $n$ letters: $$\epsilon: S_n \to \{-1,+1\}$$ $$\epsilon(\sigma) = \prod_{1 \le i < j \le n} ...
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70 views

Visualising Group Theory

I am looking for how and why people use triangular graph paper for group theory? I have read a book that used it a year or so ago but it didn't develop into anything significant. An example is this ...
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42 views

Is the following proof of $Z(S_n)=\{id\}$ correct?

I wanted to prove that the center of the symmetric group $S_n$, $n\geq 3$ is trivial. Is my argument correct? Suppose $\alpha\in Z(S_n)$, that is $\alpha\beta=\beta\alpha$ for all $\beta\in S_n$. We ...
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43 views

How to calculate $a_5,a_4=$

Let $a_n$ be the number of those permutation $\sigma $ on $\{1,2,3...n\}$ such that $\sigma $ is a product of exactly two disjoint cycles .Then $a_5,a_4=?$ Calculating $a_4$ :Possible cases which ...
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2answers
64 views

Dummit and Foote exercise verification?

I was working on the following problem: Let $\sigma$ be the m-cycle $(1 2...m)$. Show that $\sigma^{i}$ is also an m-cycle iff $\gcd(i,m)=1$ A solution to this problem is given here. But the ...
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1answer
16 views

Cycle decompositions of $S_{4}$?

Can someone please explain how to form the cycle decompositions of $S_{4}$, more specifically those that have order 2? I understand where $(1 2),(1 3),(1 4),(2 3),(2 4),(3 4)$, come from. But I ...
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2answers
32 views

How many disjoint product of two $2$-cylces are there in $S_5$ ?

How many disjoint product of two $2$-cylces are there in $S_5$ ? In general I'm having trouble in determining no. of disjoint product of cylces . Please help . Thanks in advance
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2answers
38 views

To find order of permutation

Let $\sigma$ be the permutation given by Is their a short way to do this.Thanks
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4answers
85 views

Alternative way to show that a simple group of order $60$ can not have a cyclic subgroup of order $6$

Suppose $G$ is a simple group of order $60$, show that $G$ can not have a subgroup isomorphic to $ \frac {\bf Z}{6 \bf Z}$. Of course, one way to do this is to note that only simple group of ...
8
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2answers
247 views

What does it mean when people say that groups are a study of symmetry?

I see many people make remarks to the effect that groups have basic symmetry properties. I am familiar with Cayley's Theorem and the symmetric groups $S_N$. However, when I think of the symmetric ...
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0answers
14 views

How does the weighted superposition of irreps make the Fourier transform of a finite group unitary?

This is supplementary to this question. In the lecture note of Andrew Childs on Nonabelian Fourier analysis, it is said that the Fourier transform of a finite group is the weighted superposition of ...