0
votes
0answers
13 views

Compounding unary operators

I am working with the symmetric group $S_5$. I have 3 unary operators defined: $R$, $T$, and $O$, and I'm writing about their composition. Suppose I want to denote the compound operation of "$T$, ...
0
votes
1answer
80 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
1
vote
6answers
67 views

Generators of the Symmetric group $S_3$

I am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not found ...
2
votes
1answer
53 views

$GL_2(R)$ and $\mathbb{A}_4$

So, i have to prove that $\mathbb{A}_4$ is not isomorphic to a subgroup $G$ of $GL_2(R)$. Here is a "hint" (are previuos part of the same problem that i was thinking, so i supose should help): If ...
5
votes
2answers
58 views

Subgroups of order $5$ in Icosahedral group

Let $G$ denote the orientation preserving isometries of Icosahedron. I want to show the following using group-theoretic notions: Let $N\leq G$ be a subgroup with order $5.$ Show that it is a ...
3
votes
1answer
48 views

Show that Icosahedral group does not have normal subgroup of order 5

Here is my work so far: Let $G$ denote the group of orientation-preserving isometries of Icosahedron. I have shown that $G$ acts transitively on $G/G_s$ where $G_s$ is the isotropy group, namely ...
0
votes
3answers
52 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 ...
0
votes
1answer
22 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 ...
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
44 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
27 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 ...
4
votes
0answers
51 views

largest permutation group of odd order

For general $n$, what is the largest subgroup of the symmetric group $S_n$ that has odd order? I have a feeling that it might be the Sylow 3-subgroup...ADDED: but it isn't, as Mark Bennet points out ...
2
votes
2answers
73 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
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 ...
2
votes
2answers
58 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
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
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 ...
0
votes
1answer
46 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 ...
0
votes
2answers
34 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
56 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
87 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
32 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
123 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. $without$ $theory!$
1
vote
2answers
100 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.
0
votes
1answer
36 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. ...
3
votes
2answers
136 views

Proving that any permutation in $S_n$ can be written as a product of disjoint cycles

I have attempted a proof of this, but upon looking at my notes, I feel it might be incorrect: it is noticeably simpler than the one in my notes. Proposition: any permutation in $S_n$ can be written ...
1
vote
0answers
90 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) ...
1
vote
3answers
104 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 ...
1
vote
1answer
60 views

Let $\tau_1, \tau_2 \in A_6$ be permutations with cycle-type $(2, 4)$. Show there exists $\tau \in A_6: \tau \tau_1 \tau^{-1} = \tau_2.$

Let $\tau_1, \tau_2 \in A_6$ be permutations with cycle-type $(2, 4)$. Show there exists $$\tau \in A_6\mid \tau \tau_1 \tau^{-1} = \tau_2$$ I know $\tau_1 = (a b c d)(e f)$ and $\tau_2 = ...
4
votes
1answer
80 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 ...
2
votes
2answers
94 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 ...
2
votes
2answers
92 views

Find the number of permutations in $S_n$ containing fixed elements in one cycle

Find the number of permutations in $S_n$ such as 1 and 2 belong to one cycle.
2
votes
1answer
48 views

No group has commutator isomorphic to S_4

I'm wondering about an alternative solution to the following question from Dummit and Foote: Show that no group has commutator subgroup isomorphic to $S_4$. The hint says to use the previous ...
3
votes
4answers
166 views

Ways of expressing permutations as products of transpositions

Determine whether the following permutation is even or odd and write it as a product of transpositions in two different ways. $(1527)(3567)(273)$ So far, I have the following: ...
3
votes
1answer
73 views

How to show that two groups makes $S_n$

I need to show that: $S=\left\{(12),(13),...,(1n)\right\}$ generates $S_n$ $S=\left\{(12),(123\cdots n)\right\}$ generates $S_n$ How do I show that each one of them generates $S_n$? Thank you!
1
vote
0answers
93 views

Alternating group $\operatorname{Alt}(n)$, $n>4$ has no subgroup of index less than $n$.

I have read somewhere (do know where) the following statement. Alternating group $\operatorname{Alt}(n)$, $n>4$ has no subgroup of index less than $n$. I want to prove it. If there is a ...
7
votes
0answers
95 views

A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? ...
1
vote
1answer
127 views

Operation table for the quotient group $S_4/V$

So I need to write out an operation table for the quotient group $S_4/V$, where $$V = \{(e),(12)(34),(13)(24),(14)(23)\}$$ thus $|V|$ = 4 and $|S_4|$ = 24. My question is: Do I need to really write ...
1
vote
1answer
96 views

If a subgroup of a symmetric group has an odd permutation, then it has a subgroup of index 2.

I want to show that for $n\geq 2$ and $H\leq S_n$ if $H$ contains an odd permutation then it necessarily has a subgroup of index 2. I am not sure how to start, if it has an odd permutation then it ...
2
votes
1answer
42 views

Center of $D_6$ is $\mathbb{Z}_2$

The center of $D_6$ is isomorphic to $\mathbb{Z}_2$. I have that $$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ ...
1
vote
3answers
96 views

Relationship between group actions and homomorphisms

I know that there exist no nontrivial homomorphism from $S_3$ into $Z_5$ as they are groups of co-prime order. I am not looking for an explanation of this but for an explanation concerning the obvious ...
0
votes
0answers
33 views

The group generated by all permutations of an orthonormal basis and their negatives

Consider the group generated by all permutations of an orthonormal basis in $\mathbb{R}^n$, and by taking the negatives of these basis vectors. The general element of this group is a matrix (in the ...
3
votes
1answer
52 views

How to show that $S_4=A_3D_4$ and $S_4=A_3K$?

Ok, so I know how to do this by direct computation, but that seems like it will take a long time, and that is probably not how my professor wants this done. This is a practice question for the exam, ...
3
votes
1answer
135 views

Why is conjugation by an odd permutation in $S_n$ not an inner automorphism on $A_n$?

I was reading about the outer automorphism group on wikipedia, and it mentions that conjugation by an odd permutation is an outer automorphism on the alternating group $A_n$. This suggests the ...
4
votes
3answers
138 views

Proving that two permutation groups are isomorphic

Here's the statement to prove: Let $n,m$ be two positive integers with $m≤n$. Prove that $S_m$ is isomorphic to a subgroup of $S_n$, where $S_n$ is the collection of all permutations of the set ...
1
vote
0answers
51 views

Index of centralizer of alternating group

For any element $x \in A_5$, we have that $$[A_5:C_{A_5}(x)]=\begin{cases} [S_5:C_{S_5}(x)], & \text{condition 1} \\ \frac{1}{2}[S_5:C_{S_5}(x)], & \text{condition 2} \end{cases}$$ Basically, ...
1
vote
1answer
283 views

4 Element abelian subgroup of S5.

I have a homework question from my intro to group theory class. Question: Find a 4 element abelian subgroup of $S_5$. Write it's table. This is where I've gotten so far, but I don't even know if ...
1
vote
2answers
129 views

Which elements of $S_8$ are in the subgroup of rigid motions of a cube?

Let the set $S\colon= \{ 1, 2, 3, 4, 5, 6, 7, 8 \}$. Then which permutations of $S$ will appear in the group of rigid motions of a cube, which is a subgroup of $S_8$, the symmetric group on 8 letters? ...
3
votes
1answer
249 views

Without Lagrange's theorem: If $H$ is a subgroup of odd order of $S_n$, then it is a subgroup of $A_n$.

Prove the following without Lagrange's theorem: If $H$ is a subgroup of odd order of $S_n$, then it is a subgroup of $A_n$. So this proof is pretty trivial if you have Lagrange's theorem, but ...
4
votes
1answer
76 views

Necessary and Sufficient conditions to generate $S_n$

I have a homework question that asks "Find necessary and sufficient conditions on $1 \leq i < j \leq n$ so that $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$." Here is what I have done ...