# Tagged Questions

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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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!
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### Center of $D_6$ is $\mathbb{Z}_2$

The center of $D_6$ is isomorphic to $\mathbb{Z}_2$. I have that $$D_6=\left< a,b \mid a^6=b^2=e,\, ba=a^{-1}b\right>$$ $$\Rightarrow D_6=\{e,a,a^2,a^3,a^4,a^5,b,ab,a^2b,a^3b,a^4b,a^5b\}.$$ ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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? ...
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### 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 ...
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### 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 ...
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### $(1 2)(3 4)$ does not commute with any nonidentity element of odd order in $A_5$.

On Dummit's Abstract Algebra on p. 128, it says: "It is easy to see that $(1 2)(3 4)$ ... does not commute with any non-identity element of odd order in $A_5$." But I don't find it easy. Any ...
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### Chromatic classes of vertices of a polyhedron

For a convex polyhedron, how do I figure out all possible proper chromatic classes of its vertices (so that all vertices that are assigned the same color constitute a separate class, and no two ...
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### Conjugate subgroups and conjugate elements

While trying to prove that the alternating group $A_5$ is a simple group, I came across two assertions I see as contradicting, that is : the 5-cycles are not all conjugate to each other (proven ...
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### A problem of permutation group

An exercise in a book of permutation groups: Let $G \leq S_n$. If $G$ has $r$ orbits, show that $G$ can be generated by a set of at most $n-r$ elements. I really have no idea how to prove it. ...
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### Algebraic expressions and permutation groups

Suppose that I pick a subgroup $G$ of $S_n$ for some $n$. Is it always possible to find an algebraic expression in $n$ variables (in other words, a rational function in those $n$ variables) that is ...
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### Proving that doesn't exist subgroup H with order 6 in $A_4$ [duplicate]

Let $G=A_4$. Prove that does not exist subgroup $H\le G$ s.t $|H|=6$. I don't know from where to start (maybe I need to prove that if so $H\triangleleft A_4$?) Any hint will be appreciated.
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### Find all numbers $n$ such that $S_7$ contains an element of order $n.$

Find all numbers $n$ such that $S_7$ contains an element of order $n.$ Identity is the only element of order $1.$So $n=1$ is possible. Case 1: Elements that can be written as a unique cycle of ...
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### Simple subgroup of Symmetric Group

I have the following question: Let $n\geq5$, and suppose that $G$ is a simple subgroup of $S_{n+1}$ of index $k$. Show that if $k\leq2n+2$, then $G=A_{n+1}$ or $G$ is isomorphic to $A_n$. I have ...
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### The structure of a group of order 443520

Let $G$ be a finit group and $T$ be a normal subgroup of $G$ such that $PSL(3,4) \unlhd T \leq Aut(PSL(3,4))$ and $|T|=2|PSL(3,4)|$. If $G= T\rtimes C_{11}$, then what we can say about the structure ...
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### Definition of the Young Symmetrizer

I have a question regarding the equivalence of two definitions of the young symmetrizer. First, some notation: let $\lambda$ be a partition of $n$. Given a $\lambda$-tableaux $T$, we define the row ...
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### Order of elements in $S_4$

Let $r(n) = \left| \left\{ \sigma \in S_4 : \mbox{ord} ( \sigma) = n \right\} \right|$. Is it true that: $r(2)>r(4)$ $r(4) > r(3)$ $r(1)+r(3) = r(2)$ $r(5) = r(6)$ I can write all elements ...
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### What is the smallest $n$ for which the usual “counting sizes of conjugacy classes” proof of simplicity fails for $A_n$?

A common proof of the simplicity of $A_5$ proceeds as follows. First, note that a normal subgroup is always a union of conjugacy classes. Also, a subgroup has order dividing the size of the group by ...
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### Conjugacy Class of symmetry group $S_{10}$

Let $X=\{a\in S_{10} | ~~\text{order}(a)=8\}$. Determine how many conjugacy classes are in $X$. How to do this question in general?
For symmetric group $S_n$, we need to find a collection of subgroups $G_i$'s such that union of these subgroups is the group $S_n$ and each subgroup found is isomorphic to direct product of cyclic ...