3
votes
2answers
62 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
134 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
35 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 $$\phi(a) =\lambda_a(x)=ax \text{ where } x\in S $$ ...
0
votes
0answers
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
votes
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
votes
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 ...
0
votes
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 ...
1
vote
0answers
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?
0
votes
1answer
29 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
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 ...
1
vote
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 ...
1
vote
1answer
56 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
62 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
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, $(1 2)^{(1 2 3 \ldots n)}=(1 3)$.
0
votes
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, ...
1
vote
2answers
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 ...
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
85 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
51 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
48 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
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
votes
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 ...
5
votes
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 ...
3
votes
1answer
78 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 ...
2
votes
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 ...
2
votes
0answers
52 views

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 $n$ ...
1
vote
0answers
39 views

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
votes
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 ...
0
votes
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 ...
0
votes
2answers
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 $S_4$.
0
votes
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 ...
2
votes
2answers
29 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.
1
vote
5answers
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. $without$ $theory!$
0
votes
1answer
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 ...
10
votes
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 ...
0
votes
2answers
92 views

Find all the elements in $S_4$ that commutes with $(12)(34)$.

Find all the elements in $S_4$ that commutes with $(12)(34)$. And show the algorithm of the process.
5
votes
1answer
118 views

For $n \ge 3$, the only central element of $S_n$ is the identity - Fraleigh p. 86 8.47

Strengthening Exercise 46, show that the only element of $S_{\large{n \ge 3}}$ satisfying $\sigma\gamma = \gamma\sigma$ for all $\gamma \in S_n$ is the identity permutation. I call this $i$. The ...
0
votes
1answer
32 views

If $\sigma\in S_n$ is of length $n$, then $\sigma$ generates its centralizer

Let $S_n$ and $C_G(\sigma)$ denote the symmetric goup and the centralizer of $\sigma\in S_n$, respectively. I want to show: If $\sigma$ is of length $n$, then $C_G(\sigma)=\langle\sigma\rangle$, i.e. ...
1
vote
1answer
47 views

A group containing a copy of $S_n$ for $n = 1,2,\cdots$

Suppose that $G$ is a group containing a copy of $S_n$ for each $n$. Does it mean that $G$ contains a copy of $S_\Bbb{N}$?
1
vote
1answer
37 views

Does the isomorphic image of a stabilizer subgroup fix a point

Here is an exercise from [Birkhoff and MacLane, A Survey of Modern Algebra]: Let $\phi: G \rightarrow G'$ be an isomorphism between two groups of permutations. Let $S$ consist of those ...
2
votes
3answers
76 views

maximum order of an element in symmetric group [duplicate]

While doing my homework i find out that the maximum order of an element in $S_3$ is 3 (the element $(123)$) and the maximum order of an element in $S_4$ is 4 (the element $(1234)$) Can i generalize ...
0
votes
3answers
223 views

Find a set of generators and relations for $S_3$

Can you help me with this? By trial, I came up with the generators and relation. However, how do I prove that the generators and relations uniquely determine $S_3$? Problem Find a set of generators ...
3
votes
1answer
68 views

Alternating group generators

Consider the alternating group $\mathcal A_n$ ($n$ is an odd integer). Do $(12\cdots n)$ and $(12)(34)$ generate $\mathcal A_n$? In other words, $\langle (12\cdots n),(12)(34)\rangle =\mathcal A_n$. ...
1
vote
0answers
65 views

Decomposition of representation of symmetric group

Let $V$, $\dim V=n-1$ be the standard representation of the symmetric group $S_n$ and let $V'= \langle x_1,x_2,\ldots,x_n \rangle$ be its natural representation. Then ( see. Fulton, Harris, 4.19) ...
0
votes
0answers
14 views

Help with finding some interesting rotation group of cube

See this site here : http://3.cnxical.appspot.com The cube has 6 faces. Clicking on each face applies a group element that moves the cube through rotations about the X and Y axes. I started with ...
1
vote
3answers
83 views

Subgroup of a symmetric group $S_7$

Is there any quick way to determine if $S_7$ contains a subgroup of order $6$ or $S_{11}$ a subgroup of order $30$ ? A problem carrying only 2 marks involves this. So I assume either there is some ...
4
votes
1answer
74 views

What's the smallest $n$ such that $D_{12}$ has an isomorphic copy in $S_n$?

I can show that $S_7$ is the smallest candidate for the property given. And, with a little calculation, I think it works out - $(1234)(567)$ and $(14)(23)(57)$ seem to generate such a subgroup. But I ...
6
votes
2answers
86 views

structure of the full symmetric group on a countably infinite set

trying to get a handle on the full symmetric group $S$ of permutations on a countable set $X$. i had never really thought much about this group, but now i look at it for the first time it appears a ...
2
votes
1answer
66 views

$\sigma$ = (1 2 3 4 5). Find a $\gamma$ in S5 such that:

(a) $\gamma$ $\sigma$ $\gamma^{-1}$=$\sigma^2$ (b) $\gamma$ $\sigma$ $\gamma^{-1}$=$\sigma^{-1}$ (c) $\gamma$ $\sigma$ $\gamma^{-1}$=$\sigma^{-2}$. I thought that there was a theorem that stated ...
2
votes
2answers
81 views

Intuitive idea on generators of $S_4$

What is the way to convince myself that $\left\langle(1,2),\ (1,2,3,4)\right\rangle=S_4$ but $\left\langle(1,3),\ (1,2,3,4)\right\rangle\ne S_4$? Let $\sigma$ be any transposition and $\tau$ be ...