# Tagged Questions

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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### How many elements in $S_{8}$ are conjugate with $(12)(345)$?

How many elements in $S_{8}$ are conjugate with $(12)(345)$? My reasoning is as follows: Two elements in $S_n$ are conjugate if and only if they have the same cycle type, so we need to count the ...
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### Sylow subgroup of a symmetric group

Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which ...
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### Difference between symmetry algebra and symmetry group

What is the difference between symmetry algebra and a symmetry group? I just wanted to know if my understanding is right. Lets say we have a system of differential equations. Then the symmetry group ...
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### If $A$ is a matrix with negative eigenvalues, then $\exists M$ : $A = -MM^T$

Let $A$ be a symmetric matrix with all its eigenvalues negative. Prove that there exists a matrix $M$ such that : $A = -MM^T$. Now, regarding my question, I have found another older question, that ...
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### $(C_2 )^3$ is not a subgroup of $S_4$

Prove $(C_2)^3$ is not a subgroup of $S_4$. (Using group actions.) I could think of a permutation argument that $(C_2)^3$ is not a subgroup of $S_4$. But I would like to argue it by considering ...
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### Show that the product of two transpositions can be expressed as a product of $3$-cycles

Consider the symmetric group $S_n$ where $n>2.$ Show that the product of two transpositions $(ab),\,(cd)$ can be written as a product of $3$-cycles where $a,b,c,d$ are all distinct. I'm not sure ...
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### The number of Sylow $5$-subgroups in $S_6$

Find the number of sylow $5$-subgroups in $S_6$. First: $ord(S_6)=6!=2^4\cdot 3^2\cdot 5=144\cdot 5$, so $n_5|144$ and $n_5\equiv 1\pmod5$, where $n_5$ is the number of sylow $5$-subgroups. Since ...
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### Show that a set is a right transversal for $S_3$ in $S_4$

I am given the set A ={e,(14),(24),(34)} I'm supposed to show that this set is a right transversal for S3 in S4, meaning that every right coset of S3 contains exactly one element of A. I'm getting ...
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### Show that if $K$ is a normal subgroup of $H$ such that $H/K$ is abelian, then K contains all of the 3-cycles.

Claim: For $n≥5$, if $H$ is a subgroup of $S_n$ which contains all of the 3-cycles, and $K$ is a normal subgroup of $H$ such that $H/K$ is abelian, then K also contains all of the 3-cycles. Attempt: ...
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### Dimension of induced representation in $S_3$

Let $G=S_3$. It has 3 irreducible representations: $1, sgn$ and $V$; the trivial rep, sign rep and rep $V$ where $dimV=2$ Consider the subgroup $H=S_2$ with irreps $1_H$ and $sgn_H$ What is the ...
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### The order of the normalizer of a $p$-subgroup of $S_{p}$ [closed]

I found it In Exercise in abstract algebra by Dummit and Foote. Let $P$ be a Sylow $p$-group of $S_p$. What is the order of $N_{S_p}(P)$?
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### About transitive subgroups of symmetric group $S_n$

When I am studying Galois theory I came across some problems: Let $S_n$ be the symmetric group on $n$ letters($|S_n|=n!$).How to determine all the transitive group $G$ of $S_n$ ( A subgroup $G$ ...
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### Ordering of elements in the base of a group

In section 4.6.7 of HANDBOOK OF COMPUTATIONAL GROUP THEORY, the authors use an ordering $\prec$ for the elements in a coset. That ordering, $\prec$, was defined in section 4.6 as follows. Throughout ...
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### Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...
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### Permutation and Linear Representation of Finite Group

By a permutation representation of a finite group $G$, we mean a homomorphism from $G$ to $S_n$, the (full) permutation group on $n$ letters. By a linear representation of a finite group $G$, we mean ...
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### How to determine whether or not $S_{n}$ contains a subgroup of order $m$

I'm currently going through group theory practice problems and I needed some assistance with the following exercise: Does $S_{9}$ have a subgroup of order $25$? Is it correct to just use Lagrange's ...
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### Order of subgroup of symmetric group

Let $X$ be a finite set, i.e. that $|X| = n$, and let $G = \operatorname{Sym}(X)$ be the symmetric group on $X$. Let $Y \subseteq X$ be a subset of $X$ and define the subset $G_Y \subseteq G$ to be ...
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### Show that the permutation $(1 \space 2 \space 3)$ can not be a cube of any element of $S_n.$

Here is my try: If there exists $a \in S_n$ such that $a^3=(1 \space 2 \space 3)$, then $a^9=e$ where $e$ is identity in $S_n$. Then $o(a)=9$. I don't know how to proceed further. Can anyone ...
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### Dense and turbulent orbits

In their 2006 paper "Turbulence, amalgamation, and generic automorphisms of homogeneous structures" Kechris and Rosendal (see here for the arXiv version of the paper) state the following proposition ...
### Counting permutations in $S_n$ with $1,2,..,k$ all in same cycle
The number of permutations in $S_n$ for which the first $k$ items $1,2,...,k$ are all in the same cycle can be shown (by a somewhat tedious argument) to be $n!/k.$ I'm looking for less computational ...
### A space with “interchangeable” coordinates, $\mathbb{R}^n / S_n$
(I'll apologize in advance for the lack of rigour in this question, I'm something of an armchair mathematician at the moment, but I do try my best): I have a space that is similar to $\mathbb R^n$ ...