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Questions tagged [symmetric-groups]

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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G has 4-Sylow 3-subgroups

I am working on the following problem: Prove that if $\lvert G \rvert = 12$ and $G$ has $4$ Sylow 3-subgroups, then $G \equiv A_4$. If you let $G$ acts by conjugation on the set containing the $4$ ...
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36 views

Permutation groups: $S_4$ and $D_4$.

Question. Determine the subgroup of $S_4$ generated by $\sigma=(1\ 2\ 3\ 4)$ e $\tau = (2\ 4)$. Show that $\left<\sigma, \tau\right> <S_4$ is isomorphic to the group of square symmetries. ...
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17 views

How to check if permutation induces an element of the Galois group.

Let $f \in \mathbb Q[X]$ be irreducible of degree $n$ with zeros $\alpha_1,\dots,\alpha_n \in \mathbb C$. Further, let $L$ be the splitting field of $f$ and $\sigma \in S_n$. Is there an easy way to ...
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1answer
34 views

One of the conjugacy classes of A4

When I was computing the conjugacy classes of $A_4$, I found that $\{(12)(34), (13)(24),(14)(23)\}$ is one of the conjugacy classes of $A_4$. This is not hard to see as the center of $A_4$ is trivial. ...
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0answers
18 views

How many graph automorphisms does the follong graph have?

Let's create a graph with 8 vertices by considering the edges of a cube and adding 2 further edges being the parallel diagonals of 2 opposite sides. How many graph automorphisms does it have? I was ...
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3answers
52 views

Let $n$ be an odd positive integer and $a\in S_n$ be an $n$-cycle. Show that the order of $C(a)$ must be odd.

I am working on the following problem from group theory: If $n$ is odd and $a\in S_n$ is an $n$-cycle, $a=(a_1,a_2,......,a_n)$, show that no element of the centralizer $C(a)=\{g\in S_n \mid ga=ag\}...
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Counting permutations in $S_n$ which take an n-cycle to another n-cycle under conjugation

Given two fixed $n$-cycles in $S_n$, $\alpha$ and $\beta$, I want to find how many unique $\sigma \in S_n$ satisfy $$\sigma \alpha \sigma^{-1} = \beta$$ In class, we have already learned that ...
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1answer
27 views

Let $G=S_n$ and fix $i \in A=\{1,2,\ldots,n\}$ and let $G_{i}=\{\alpha \in G | \alpha(i)= i\}$. Prove $G_{i} \leq G$ and find $|G_i|$

I am not sure about $|G_i|$ but I was able to show $|G_i|$ as a subgroup as follow: Define $\; \;$ $\cdot$ $: G\times A \to A$ as $\alpha.i= \alpha(i) \; \; \forall \alpha \in G, \; i\in A$ Then $\...
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1answer
52 views

Show that $\phi(g)$ is an even permutation

Let $G$ be a group of order $n$ then $G$ is isomorphic to a subgroup of $S_n$,Denote it by $\phi$, $\phi:G\to S_n$ be an monomorphism of Cayleys Theorem Let $g\in G$ has order $k$.Show that $\phi(...
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2answers
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Finding the centraliser of an element in $S_5$

Let $x = (12)(34) \in S_5$. Is there a quick way to find $C_{S_5}(x)$ ? I know that conjugation leaves cycle types unchanged. Thus, if $\sigma \in C_{S_5}(x)$, then $(12)(34) = \sigma^{-1}(12)(34)\...
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1answer
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find all $x \in S_5$ such that $x^3 = (12)$

So I have to find all $x \in S_5$ such that $x^3 = (12)$. For example, one solution would be $(12)$ itself, because its order is $2$. How can I find all of the solutions though? Is it just trial and ...
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1answer
37 views

$7$ Sylow subgroup of $A_{20}$ is not normal and $7$ Sylow subgroup of $S_{20}$ contained in $A_{20}$.

Let $H$ be a 7 Sylow subgroup of $A_{20}.$ Then prove that $H$ is not normal in $A_{20}$ and any 7 Sylow subgroup of $S_{20}$ is contained in $A_{20}.$ Proof: since $7^2$ divides order of $A_{20}$ ...
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0answers
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How many $\alpha \in S_n$ are there with $\alpha^2 = 1$? [duplicate]

How many $\alpha \in S_n$ are there with $\alpha^2 = 1$? I want to use just linear algebra, I have done it using combinatorics but in class that does not work, can I get a little help? What I know is ...
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1answer
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Express (1,…,n) as a product of 2-cycles

In Alan F. Beardon's "Algebra and geometry" he asks in an exercise to express $(1\ \ldots\ n)$ as a product of two cycles: Show that $(1\ 2\ 3\ 4)=(1\ 4)(1\ 3)(1\ 2)$. Express $(1\ 2\ 3\ 4\ 5)$ as ...
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4answers
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Equational laws holding in the symmetric group $S_3$

I'm engaged in group theory (at least I am trying to get better) and so I found a problem dealing with the symmetric group $S_3$. The first question is to find an (equational) law $\gamma$, which ...
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3answers
52 views

Group actions and orbits

Let $X=\{\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\}\}$ and $G=S_4$ (a) Show that $\phi \{x_1,x_2\}=\{\phi (x_1),\phi (x_2)\}$ determines an action of $S_4$ on $X$, where $\phi \in G$ (b) ...
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1answer
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If $\operatorname{Sym}_X \simeq \operatorname{Sym}_Y$ then there is bijection from $X \to Y$

If $\operatorname{Sym}_X \simeq \operatorname{Sym}_Y$ then there is bijection from $X \to Y$ ? , I proved the other way around, i think i need to build $f$ from $\psi : \operatorname{Sym}_X \to\...
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0answers
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Number of one one functions with no fixed points.

Let $f:A \to A$ be a function from a set $A$ with $n$ elements. How many $f$ are there which are bijective and satisfies the property $f(a)\neq a$ for any $a \in A$ ?
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1answer
27 views

Symmetry group of equilateral triangle

I have read at some places that the symmetry of equilateral triangle is C3v as well as some places mention it to be D3. The group tables for these two groups differ, hence they are not isomorphic. ...
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0answers
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Rotational symmetries of a double tetrahedron

Consider the following problem: given two copies of a regular tetrahedron, glue these two copies together along a face. Describe the rotational symmetries of the resulting solid. My approach. The ...
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1answer
38 views

Lie algebra generated by elements of the symmetric group $S_N$

For an arbitrary group $G$, the group algebra $\mathbb{C}(G)$ is defined as the set of all formal linear combinations of the elements of $G$: $\mathbb{C}(G)=\{c_1g_1+c_2g_2+\ldots+c_n g_n|c_i\in \...
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41 views

Power of a r-cycle

Given $n \ge 3$ and $2 \le r \le n$, let's define the permutation $\rho \in S_n$ (the symmetric group of degree $n$) by: $\rho(k)=k+1$, for $k \in I_<:=\lbrace 1,...,r-1 \rbrace$ $\rho(r)=1$ $\rho(...
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1answer
84 views

Schur-Weyl duality for qubits

I am interested in applying Schur-Weyl duality to quantum information theory, specifically "qubits". But I have been stuck for some time on understanding how the Young symmetrizers work in this ...
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2answers
51 views

Suppose a pizza has 4 slices, and each slice can be topped with either peppers, onions, or both. How many different pizzas can be made?

Suppose a pizza has 4 slices, and each slice can be topped with either peppers, onions, or both. How many different pizzas can be made? I am using Burnside's counting theorem with the group $G=\left \...
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1answer
27 views

Order of the center of $Sym(n)$ [duplicate]

With the standard notations, what's $|Z(Sym(n))|$ equal to, in terms of $n$?
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1answer
41 views

number of orbits of $A_5$ acting by left multiplication in $S_5$

Looking for a very fast/"smart" way to compute this number (it was a question asked on an hour-long exam I recently took, so listing everything out for each element in $S_5$ was not an option since I ...
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1answer
23 views

Group homomorphism from $A_n$ to $C^*$

I need to find the the possible homomorphisms from $A_n$ (Alternating group) for $n \geq 5$ to $C^*$ (the group of non-zero complex numbers). My take: Fact: $A_n$ for $n \geq 5$ is simple For ...
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0answers
55 views

The automorphism group of $S_6$ is isomorphic to a semidirect prodct

On this document an outer automorphism of $S_6$ is constructed. I would like to use this construction to prove that $\mathrm{Aut}(S_6)\cong S_6\rtimes_\varphi\mathbb{Z}_2$. The idea would be to find ...
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1answer
24 views

Prove that the alternating group $A_{n}$ is generated by all products of two 2-cycles.

Prove that the alternating group $A_{n}$ is generated by all products of two 2-cycles ($i$,$j$)($k$,$l$). These 2-cycles are not necessarily disjoint. So I know that $A_{n}$ can be written as ...
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Generating Symmetric Group from a subset

Show that Sn (the finite symmetric group) is generated by the subset {(1, 2),(2, 3),...,(n − 1, n)} by using the formula σ·(a1,...,ak)·σ−1 = (σ(a1),...,σ(ak)) I know that we can get any k-cycle from ...
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0answers
171 views

Is this proof on the Sylow 5-subgroups of G 100% correct?

Problem statement : Let $G=S_5$ (it actually reads $|G|=S_5$ but this must be a misprint obviously). Using Sylow , how many 5-Sylow subgroups has G got? Is this consistent with the number of 5 cycles ...
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1answer
36 views

A question on $G=S_5$ concerning notation

I think there must be a misprint in a question I've been asked.It has written on it ; $$|G|=S_5$$ but of course this notation is nonsense as a real number cant be equal to a set. Do you think it ...
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1answer
20 views

Conjugacy in Galois group and in symmetric group

Let $f \in \Bbb Q[X]$ be an irreducible polynomial with complex roots $a_1, .., a_n$ and splitting field $K$. The action of $G:=Gal(K/\Bbb Q)$ on the roots gives an embedding $\rho : G \to S_n$. If $\...
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1answer
52 views

Prove that $S_4 \cong V_4 \rtimes_\phi S_3$ for any isomorphism $\phi: S_3 \to \text{Aut}(V_4)$

Note that $\text{Aut}(V_4) \cong S_3$. I know how to prove that $S_4$ isomorphic to some semidirect product of $V_4$ and $S_3$. I know if it works for an isomphorism it works for any isomorphism. ...
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2answers
72 views

Why are there 48 symmetries of a cube?

I'm trying to prove that there are a total of 24 rotation and 24 reflection symmetries of a cube. I can show the first part, but I don't have a good proof for why there are also 24 reflections. The ...
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1answer
25 views

$Hol(\Bbb Z_2 \times \Bbb Z_2)$

How to find $Hol(\Bbb Z_2 \times \Bbb Z_2)$? where $Hol(\Bbb Z_2 \times \Bbb Z_2)=(\Bbb Z_2 \times \Bbb Z_2)\rtimes_{id} Aut(\Bbb Z_2 \times \Bbb Z_2)$. Now we know that $Aut(\Bbb Z_2 \times \Bbb Z_2)...
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1answer
40 views

Centraliser of (12)(34) in A5

In the proof that A5 is simple, Dummit and Foote p.128 claim that (1) it is easy to see that (12)(34) commutes with (13)(24) but does not commute with any element of odd order in A5. And further (2) ...
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1answer
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how many cycles of each type in S6

how many cycles of each type in S6? I know I can write down all cycles. I wonder is there a formula for me to quickly calculate the number of cycles of each type in Sn?
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Show that $σ(1\, 2\, 3\, 4 … k)σ^{−1}=(σ(1)\,σ(2)…σ(k))$, where $(σ(1)\,σ(2)…σ(k))$ is another k-cycle.

Let $σ∈S_n$ and $(1 2 3 4 ... k)∈S_n$ be a $k$-cycle. Show that $σ(1\,2\, 3\, 4\, ... k)σ^{−1}=(σ(1)\,σ(2)…σ(k))$, where $(σ(1)\,σ(2)…σ(k))$ is another k-cycle. My current thought is: We know $σ$ ...
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0answers
84 views

Types of group action observing the table of $G$ acting in $X$

I'm studying the group action of a group $G$ on a set $X=\{1,\dots,n\}$ For example taking a straightforward example with $G \leq S_4$: $$X = \{1,2,3,4\} \quad G=\{e_4,(1,2),(3,4),(1,2)(3,4)\}$$ We ...
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0answers
36 views

Confusion on Irreducible representations of a group

There are two questions which are interrelated hence I want to mention them over here. Given the irrep $\Gamma^{(3)}$ of group C3V, which is of 2 dim. It can be diagonalized further, into a simple ...
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1answer
28 views

Finding the number of elements of order 7 in $S_9$

I'm trying to find how many elements in $S_9$ have order $7$. Using the fact the order of $g$ is the lowest common multiple of the length of its disjoint cycles, the only combination of cycles ...
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1answer
32 views

Making sense of translational invariance of the heat equation

The 1-D heat equation $u_t=\gamma u_{xx}$ admits the transformation $$\frac{\partial}{\partial x}, \mbox{ translation along x}$$ The solution $u$ is invariant. This means it exhibits translational ...
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1answer
30 views

When do $(i \enspace j)$ and $(1 \enspace 2 \ldots n)$ generate $\mathfrak{S}_n$?

Let $n$ be an integer greater than $1$, and let $i, j\in \{1, \ldots ,n\}$ with $i<j$. The permutations $(i \enspace j)$ and $(1 \enspace 2 \ldots n)$ generate the whole symmetric group $\mathfrak{...
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2answers
347 views

What does order of element mean in the Symmetric Group?

I have seen several references to "order" of an element in the Symmetric Group. Specifically, that the order of a cycle is the least common multiple of the lengths of the cycles in its decomposition. ...
2
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1answer
45 views

Computing the centraliser for elements in $S_4$.

Given the group $S_4$ we know that it has 5 cycle types of type. I want to pick one element from each of the conjugacy classes and find its centralizer. $$ \begin{matrix} type & no....
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0answers
70 views

Approximation of products of necklaces

Let's consider approximation of $\prod_{p=1}^n N(p,a)$, $n\to \infty$, where the number of fixed necklaces of length n composed of $a$ types of beads $N(n,a)$: $\frac {(a-1)^{n+1}} {(a-3) \cdot n!} \...
4
votes
2answers
116 views

Showing that $H\leq S_n$ containing rotations must be isomorphic to $D_n$

Let $H$ be a subgroup of $S_n$ such that $H$ is isomorphic to the dihedral group $D_n$. Let also $K$ be a subgroup of $S_n$ such that $K$ is isomorphic of $D_n$. I would like to show that if $K$ ...
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4answers
47 views

Permutation group products (23)(12)(34)=(1243)?

I just wanted to ask why it makes sense that $(23)(12)(34)=(1243)$. Note I'm going from right to left. I'm trying to understand the concepts trying to find a true "method" to follow in all cases. My ...
0
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0answers
18 views

$A_n$-orbits in $ \frac{S_n}{D_{2n}}$

Let $X$ := $ \frac{S_n}{D_{2n}}$, I want to use the orbit-stabilizer theorem to prove that the size of the set of $A_{n}$-orbits in $X$ is $2$ if $n\equiv 1 \pmod 4$ and $1$ if $n$ is anything else. ...