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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permutation problem: cycle representation

Let $n$ be an odd number. Let $C_n$ be the set of permutations $\pi$ of $[n]$ whose cycle representation has only one cycle. Let $\pi,\sigma\in C_n$. Prove that their composition $\pi\sigma$ has an ...
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67 views

What is the reason for stating Cayley's theorem this way?

In my notes, Cayley's theorem reads: Any group $G$ is isomorphic to a subgroup of $\text{Sym}\, X$ for some $X$. On the other hand, several sources (such as Wikipedia) give a slightly more ...
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3answers
136 views

Can we find an element of infinite order in a symmetric group of infinite order?

In particular, I'm thinking of a simple example: the group $S_\Omega$ given $\Omega = \{1, 2, 3, ...\}$. I've been thinking of elements of $S_\Omega$ in terms of their cycle decomposition, which may ...
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0answers
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Jucys-Murphy elements confusion

I am taking a class called "Harmonic Analysis on Finite Groups" and am studying for an exam. We have recently been talking about the representation theory of the symmetric group (over $\mathbb C$). ...
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1answer
27 views

Finding the maximum possible order for an element in $S_5$

I understand that you have to write out all the disjoint cycles and then take the least common multiple which yields the highest order. But my question is, do I have to write all elements of $S_5$, ...
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2answers
36 views

Kernel of $\phi:G \rightarrow \operatorname{Sym}(S)$ Group actions

$\operatorname{Sym}(S) == \text{All permutations of the set }S$. Prove $\ker(\phi)=\bigcap_{x\in S}G_x$ where $G_x$ is the stabilizer of $x$. Let $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
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1answer
41 views

Finding the smallest positive integer $n$ such that $S_n$ contains an element of order 60.

I am trying to find the smallest positive integer $n$ such that $S_n$ contains an element of order 60. I know that every permutation in $S_n$ can be expressed as the product of disjoint cycles, and I ...
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38 views

Reflections in symmetric groups

Define the action of $S_n$ on ${\rm I\!R}^n$ as a permutation of the standard basis vectors $e_1,\ e_2,\ ...\ ,e_n$. For $\phi \in S_n$, $\phi (e_k) = e_{\phi (k)}$. How can I show that all the ...
2
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1answer
47 views

How did the Symmetric group and Alternating group come to be named as such?

The Dihedral group makes sense, "Di" means two, and "hedral" means.. shape I think (I've just realised how much of what I think words mean are guesses based on experience) like a "polygon" is a 2d ...
2
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2answers
53 views

How can I prove that $C_{S_4}((12)(34))$ is a subgroup of order 8?

I would like to know if there is a smart method (i.e. avoid to make all by hand) to understand the order of $C_{S_4}((12)(34))$. The only thing I thought is that ...
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3answers
50 views

Prove $S_4$ has only 1 subgroup of order 12

The subgroup in $S_4$ that I know has order 12 is the subgroup of all even permutations, otherwise known as the alternating group $A_4$. However, I know this from a fact and not because I am able to ...
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35 views

Subgroup of $S_4$ with order 12 [duplicate]

I need to find a subgroup of $S_4$ that has order 12 other than the subgroup of even permutations or anything isomorphic to it. What is the procedure to go about finding such a subgroup?
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1answer
30 views

Proof that $S_3$ is the smallest non-commutative group

Before I start attempting to show that every group of order less than 6 is commutative, is there a shorter/faster way to go about proving this?
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1answer
26 views

Given certain set of symmetries of a tensor, how do you associate the corresponding young tableaux

I have a particular problem, the following. $T^{a_1 \dots a_p;b_1 \dots b_p}$ is a tensor with the following symmetries. 1) $a_i$'s and $b_i$'s are completely antisymmetric, ie restricted to ...
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1answer
42 views

Normalizer of a Sylow subgroup

In this question Sylow subgroups of soluble groups Jack Schmidt mentions that the normalizer of $P$ in $S_p$ is solvable. Suppose $P$ is generated by a cycle of length $p$. Could you provide any ...
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2answers
74 views

Embedding $S_n$ into $A_{n+2}$

I am trying to prove that for all $n$, $S_n$ is isomorphic to a subgroup of $A_{n+2}$. Say $S_n$ acts on $\{\alpha_1,...,\alpha_n\}$ and $A_{n+2}$ acts on ...
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1answer
57 views

Generating the symmetric group $S_n$

I know that $\sigma =(1 2 \ldots n)$ and $\tau =(1 2)$ together should generate the symmetric group by virtue of conjugation, i.e. $(\sigma)^k \circ \tau \circ (\sigma^{-1})^k = (k+1, k+2)$; we know ...
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16 views

Prove Theorem 1. Now let A,B,C,D,E be the following five sets

Theorem 1. If H is a subgroup of the symmetric group S_5 with order 60, then H = A_5. The group S_5 is included in S_6 in the usual way: if σ is a permutation of {1; 2; 3; 4; 5} we extend it to a ...
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1answer
63 views

Standard represention of $S_3$

I am wondering how to extract the standard representation from the permutation representation? I want to obtain the permutation rep matrices $\Gamma((1,2)), \Gamma((1,3))$ and $\Gamma((1,3,2))$ in the ...
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4answers
100 views

Does associativity justify $(f^{-1}gf)(f^{-1}hf) = f^{-1}gff^{-1}hf$?

I'm self-studying abstract algabra (Herstein) and while working on an easy problem became uneasy with a step in my derivation. Given the symmetric group $S_n$ whose elements are bijections $f: S \to ...
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42 views

Obtaining representations of the symmetric group

Consider the following permutation representations three elements in $S_3$: $$\Gamma((1,2)) = \begin{pmatrix} 0&1&0\\1&0&0\\0&0&1 \end{pmatrix}\,\,\,\,;\Gamma((1,3)) = ...
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1answer
75 views

What does permutation stand for as a power?

I am just reading some books about abstract algebra and I don't understand what a permutation stands for as a power. For example, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
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20 views

Identity with symmetric rational functions

I am trying to prove this identity between rational functions involving symmetrization among variables. Let us consider a set of variables $\{p_1,\ldots,p_n\}$, which I indicate globally as ...
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1answer
74 views

Symmetric group is not cyclic [closed]

Show that for $n > 2$ the group $S_n$ is not cyclic, but can be generated by two elements. Guys how can I show this? Now how can I prove my problem?
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1answer
51 views

Cyclic subgroup of $S_n$

Find the smallest natural number n ∈ ℕ such that $S_n$ contains a cyclic subgroup of order 101. Proof: We seek the smallest n such that Sn contains a permutation of order 101. Permutation can be ...
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2answers
52 views

Intuition behind symmetric groups

I am having a hard time understanding the intuition behind symmetric groups, and in particular, their elements. Consider the group $S_3$, with elements $id, (1, 2), (2, 3), (1, 3), (1, 2, 3), (1, 3, ...
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57 views

Rotation Group of a Soccer Ball

I am attempting to show that a soccer ball cannot have a 60 degree rotational symmetry through a line through the centers of two opposite hexagons. My proof so far: If it had such a symmetry, let's ...
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0answers
22 views

Show that there does not exist any non-zero homomorphism f : S3 → Z3. [duplicate]

I am not the best at group theory, a lot of concepts are still a little fuzzy for me so I'd really appreciate explanations that are in detail. Currently trying to prove this using the first ...
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2answers
86 views

The path to understanding Frieze Groups

What is the "Path" for understanding what Frieze Groups really are? Generally in mathematics, there is a is a path or "building blocks" approach to learning something. For example if I know how to ...
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1answer
52 views

Listing left and right cosets

List the left cosets $(gH)$ and right cosets $(Hg)$ for $H = \langle (123) \rangle$, where $H \le G$ and $G = S_3$. My work so far: $G = S_3 = \langle (12) (13) \rangle = \{ e, (12), (13), (23), ...
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1answer
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Find homomorphisms from $C_8$ to $S_3$ and from $C_6$ to $S_3$, and are there any isomorphisms?

For $C_8$ to $S_3$: Let $C_8$ be generated by $x$. Then $C_8 = \langle x : x^8 = e \rangle = \{e,x,x^2,x^3,x^4,x^5,x^6,x^7\}$. But how do I show homormorphisms? I'm supposed to pick elements of $x$ ...
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140 views

Sylow $p$ subgroups of $S_{2p}$ and $S_{p^2}$ - Dummit foote - $4.5.45; 4.5.46$

Question is to find Sylow $p$ subgroups of $S_{2p}$ for odd prime $p$ and show that this is an abelian group of order $p^2$ Sylow $p$ subgroups of $S_{p^2}$ for odd prime $p$ and show that this is ...
2
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1answer
33 views

Embedding symmetric groups into general linear groups

Note that any finite abelian group $\Gamma$ can be embedded into $\operatorname{GL}(1,A)$ for some commutative ring $A$. Indeed, let $A=\mathbf Z[\Gamma]$ and $\Gamma\hookrightarrow ...
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2answers
102 views

What can we say about the kernel of $\phi: F_n \rightarrow S_k$

Let $F_n$ denote the free group on $n$ generators and let $S_k$ denote the symmetric group on the integers $\{1,\dots, k\}$, and the action of homomorphism $\phi$ (as given in the title) on the ...
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1answer
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Number of sylow subgroups of $A_5$ and $S_5$ - Dummit foote $4.5.31$

Question is : For $p=2,3$ and $5$ find $n_p(A_5)$ and $n_p(S_5)$. [Note that $A_4\leq A_5$] What i have done so far is : for $A_5$ we have $|A_5|=5.4.3$ possible number of sylow subgroups are ...
2
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1answer
41 views

Elements of a group

Consider the following presentation for $A_4$ $$< p, q\, |\, p^2 = (pq)^3 = q^3 =e>$$ There exist eight elements of order $3$ in this group. Deduce these elements by writing them as products ...
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On the centralizers of $n$-cycles and conjugacy in $A_n$

I'd appreciate comments on the validity of these attempted proofs. Thanks. Let $a$ be an $n$-cycle in $S_n$. a) Show that the centralizer of $a$ in $S_n$ is $\langle a \rangle$. b) Assume that $n$ ...
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Statistics for random permutations

Let $S_n$ be the symmetric group on $n$ elements, let $d$ be the Cayley distance, and let $m$ be a Haar measure on $S_n$. Let $s$ denote a random permutation with respect to $m$, i.e., $s$ is an ...
2
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3answers
36 views

Showing an Isomorphism between question group of $S_4$ and $D_6$

I have a subgroup $N$ of $S_4$, where $ N = [1, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)] $ I need to explain whether quotient group $G/N$ is isomoprhic to either $C_6$ or $D_6$ (no proof required, just an ...
2
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1answer
28 views

A question about the action of $S_n$ on $K[x_1,…,x_n]$

Let ${K}$be the field ($\,Char\,K\not=0)$. Let $n\in \mathbb{Z}^{+}$. $S_n$ acts on $K[x_1,...,x_n]$in the following way: If $p\in K[x_1,...,x_n]$ and $\sigma\in S_n$, then $\sigma p$ is the ...
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1answer
33 views

Order of a permutation using its cycle decomposition

If $A=\{1,2,...,n\}$, $\Omega _A$ is the set of all permutations over $A$, $S_n=(\Omega _A, \circ)$, then for any $\sigma \in \Omega _A$, the order $m$ of $\sigma$ (Smallest $m \in \mathbb{N}$ for ...
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2answers
31 views

In the group A/B find the order of the coset (x$_1$+2x$_3$)+B

A is an abelian free group, with the base x$_1$,x$_2$,x$_3$. will be B a sub-group that created with x$_1$+x$_2$+4x$_3$,2x$_1$-x$_2$+2x$_3$. In the group A/B find the order of the coset ...
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40 views

prove that the group of symmetry's of tetrahedron in $R^{3}$ Isomorphic to $S_4$.

prove that the group of symmetry's of tetrahedron in $R^{3}$ Isomorphic to $S_4$.
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One dimensional binary string with periodic boundaries and reflection

I have a binary string $l=(l_1,l_2,\ldots,l_{2n})$ with $l\in\{0,1\}$ and the conditions $l_i \cdot l_{i+n}=0$ for all $i$ and $\sum l_i=n$. Now, I was wondering how many distinct string exist, when a ...
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2answers
76 views

Let $\sigma \in S_n∖A_n$ ($\sigma$ not in $A_n$), prove that the order of $\sigma$ is even.

Let $\sigma \in S_n∖A_n$ ($\sigma$ not in $A_n$), prove that the order of $\sigma$ is even. I fill that i have a way to prove it: the sign of $\sigma$ is $-1$. so $(-1)^{n-t}=-1$, when $t$ is the ...
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1answer
34 views

about z(G) by concept of symplectic spaces

if G be a finite p-group and G' be isomorphic to zp, what we can say about z(G) by concept of symplectic spaces? is [G:z(G)] a perfect square? ((i take the elementary abelian group G/Z(G) as a ...
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2answers
30 views

Question about number of elements in $S_p$ and number of $p$-sylow groups.

Let $G=S_p$ where $p$ is a prime. How many elements with order $p$ in $G$, and what are they? How many $p$-sylow their is in $G$? I will be glad to see a simple solution.
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102 views

can somebody recommend a book in a group theory.

can somebody recommend a book in a group theory. that include just questions and their answers. $without$ $theory!$
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40 views

Subgroup of $S_5$ contains cycle of length $5$ and transposition

Suppose a subgroup of $S_5$ contains a cycle of length $5$ and a transposition. Must it be all of $S_5$? Say, it contains the cycle $(1 2 3 4 5)$ and a transposition $(ij)$. Then it contains a ...
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1answer
90 views

Some questions concerning the symmetric group $S_n$

Let $a_n$ be the number of permutations in $S_n$ having an square root. Is it true that $a_{2n+1} = (2n+1)a_{2n}$ ? (experimental data's shows that this is true for small values of $n$). Is there ...