4
votes
2answers
209 views

Is there a group-theoretic proof of the Riemann rearrangement theorem?

The analytic proofs of the Riemann rearrangement theorem are easy to understand but they don't explain why commutativity breaks down when you go from finite sums to infinite sums of real numbers. I ...
-2
votes
0answers
45 views

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? [on hold]

Give a examples of $p$-Sylow subgroups of $G=A_5\times S_3$ for $p=2,3,5$ and tell what group is isomorphic to each subgroup? Does there exist a subgroup of $G$ of order $180$?
0
votes
1answer
41 views

Under what conditions can the symmetric group be isomorphic to the abelian group?

The symmetric group is the set of all permutations. My question addresses the representability of the symmetric group using only additions. I am guessing that on the finite field $\mathbb{Z}/n ...
0
votes
3answers
50 views

Subgroups of Sufficiently Large Symmetric Groups / Cayley's Theorem explanation

Here's the question: is every finite group a subgroup of a symmetric group of sufficiently large order? More specifically, if a group $G$ has order $n$, then is it true that $G \le S_{n}$? For ...
1
vote
1answer
29 views

Promises in the hidden subgroup formulation of graph isomorphism problem

In the 3rd slide of the talk, Graph isomorphism, the hidden subgroup problem and identifying quantum states, by Pranab Sen, the promise of the problem is defined as follows. Given: $G$: group, ...
3
votes
1answer
26 views

Confusion about the hidden subgroup formulation of graph isomorphism

I am going through Quantum factoring, discrete logarithms and the hidden subgroup problem by Richard Jozsa. On page 13, the author discussed the hidden subgroup problem (HSP) formulation of the graph ...
0
votes
1answer
20 views

Conjugating a permutation

I am trying to see that two permutations are conjugate exactly when they have the same cycle decomposition. I fail to see that $$r(i_1,i_2,\dots,i_k)r^{−1}=(r(i_1),r(i_2),\dots,r(i_k))$$ Any thoughts ...
1
vote
1answer
39 views

Order of some $\alpha$ in $S_4$

I want to work out the order of $\alpha = \left[ \begin{align} & 1&2&&3&&4&\\&4&2&&1&&3& \end{align} \right]$ in $S_4$ Now when I think of ...
0
votes
1answer
60 views

Example of non-Abelianness of symmetric group for graphs

I know that for $n \ge 3$, $S_n$ is non-Abelian. I would like to work out an example in terms of graphs so to make it sure that I understand it right. A symmetric group of graphs of four vertices, ...
2
votes
1answer
51 views

Characterisation of Galois Group with the action of $\sigma \in S_n$ on the roots

Let $f \in K[X]$ be irreducible and separable with roots $x_1,...,x_n$ in a splitting field $L$ of $f$ over $K$. We identify $\text{Gal}(L|K)$ with $\text{Gal}(L|K)\cong G\subset S_n$. How can I see ...
2
votes
1answer
34 views

Orbit-Stabiliser Theorem applied to Symmetric group S_n

Let $G$ be the symmetric group $S_n$ acting on the n points $\{1,2,...,n\}$, let $g \in S_n$ be the n-cycle $(1,2,3,....,n)$. By applying the Orbit-Stabiliser Theorem or otherwise, prove that ...
0
votes
0answers
56 views

number of symmetries of an arbitrary graph

Given an (undirected) graph G, is there way to (approximately) estimate the order of Aut(G)-- i.e., the number of permutations ...
0
votes
2answers
53 views

Burnside's lemma - show that there are just five necklaces

Show that there are just five different necklaces which can be constructed from five white beads and three black beads. Sketch them. The lemma tells us that The number of orbits of G on X ...
3
votes
1answer
52 views

Counting all homomorphism from $S_4$ to $\mathbb{Z_6}$

I have to count number of homomorphism from $S_4$ to $\mathbb{Z_6}$. One approach that I know is by finding the possible kernel of homomorphism from $S_4$ to $\mathbb{Z_6}$. I am using another ...
2
votes
1answer
49 views

About generator of symmetric group $S_n$

I am reading this link . In Theorem $2.7 $, it is mentioned that for $n\geq 3$ except for $n = 5, 6, 8$, symmetric group $S_n$ is generated by an element of order $2$ and an element of order $3$. ...
3
votes
3answers
49 views

Conjugate to the Permutation

How many elements in $S_{12}$ are conjugate to the permutation $$\sigma=(6,2,4,8)(3,5,1)(10,11,7)(9,12)?$$ How many elements commute with $\sigma$ in $S_{12}$? I believe I use the equation ...
3
votes
0answers
40 views

Subgroups of Symmetric Group $S_4$ and Isomorphism

During my Algebra class we were given this exercise to solve at home, but I couldn't find any solution and I also did not really get the one our teacher gave us. So, the text was: Given G = S4 = ...
0
votes
0answers
43 views

Question about subgroups of a symmetric group

I have the following question: For $n\ge 5$ show that the symmetric group $S_n$ cannot have a subgroup $H$ with $3\le [S_n:H]< n$. ($[S_n:H]$ is the index of $H$ in $S_n$). This is technically ...
3
votes
1answer
26 views

Orbits under action of a subgroup on the set of conjagtes of a second subgroup

i have the following question: Let $A\leq B\leq G$ be finite groups. Then $G$ acts naturally via conjugation on the set of conjugates $A^G$. It's trivial, that there is only one orbit under this ...
11
votes
1answer
305 views

What is the mathematical significance of Penrose tiles?

I have a very limited understanding of groups and symmetry gained mostly from online videos (for eg. this one), so forgive me if this sounds ignorant. Particular parts of Penrose tilings exhibit ...
3
votes
1answer
48 views

How can I prove that $A_5$ is perfect?

I'm trying to prove that $A_5$ is perfect. The only proof I found until now is: "It follows from the fact that it's simple and non-abelian". Simplicity is quite stronger, and since I only need ...
0
votes
2answers
71 views

Does there exist a highly counterintuitive space group?

Has anyone ever proved or disproved the existence of a subgroup of the group of all distance preserving operations on R3 under the operation of composition that satisfies the following properties? ...
0
votes
1answer
24 views

tables of cyclic subgroups and conjugates

$G = S_5$, I need to construct tables for $H$ and $aHa^{-1}$ ($H =$ cyclic subgroup $(142)(35),$ and $a = (2354) \in G$) and see what can be inferred. In my attempt $H$ = $\{(142)(35), (124)(35), ...
0
votes
3answers
67 views

Normalizer of a subgroup of $GL_2(\mathbb{R})$

I have following subgroup of $GL_2(\mathbb{R}):$ $$A=\Bigg\{\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} ...
2
votes
2answers
66 views

The Cayley Representation Theorem.

This theorem states that "Any group is isomorphic to a subgroup of a group permutations." I only ask if someone could provide a simple example so that i can fully understand this theorem.
0
votes
1answer
26 views

Basic question about dimensionality of Euclidean group

I have a basic question about the dimensionality of the Euclidean group. Why are degrees of freedom greater than the dimension? I thought that a degree of freedom is the same as a dimension, as in, ...
0
votes
2answers
59 views

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 ...
3
votes
2answers
85 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 ...
2
votes
3answers
146 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 ...
2
votes
2answers
40 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 $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
2
votes
1answer
60 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
votes
2answers
57 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 ...
0
votes
3answers
70 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 ...
1
vote
0answers
37 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?
0
votes
1answer
42 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?
0
votes
1answer
52 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 ...
2
votes
2answers
84 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 ...
2
votes
1answer
84 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 ...
2
votes
1answer
67 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 ...
1
vote
1answer
79 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, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
0
votes
2answers
57 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, ...
1
vote
2answers
85 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 ...
0
votes
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 ...
1
vote
2answers
116 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 ...
0
votes
1answer
66 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), ...
2
votes
1answer
59 views

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 $x$ ...
3
votes
3answers
144 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
votes
1answer
41 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 ...
5
votes
2answers
118 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 ...
3
votes
1answer
89 views

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 ...