Questions tagged [symmetric-groups]

A symmetric group is a group consisting of all permutations of given finite set, with composition of permutations as the binary operation. Should be used with the (group-theory) tag.

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If $m<n$, show that there is a $1$-$1$ mapping $F:S_m\rightarrow S_n$ such that $F(fg)=F(f)F(g)$ for all $f,g\in S_m$

question: If $m<n$, show that there is a $1$-$1$ mapping $F:S_m\rightarrow S_n$ such that $F(fg)=F(f)F(g)$ for all $f,g\in S_m$. Where $S_n$ stands for symmetric group of degree $n$ my approach: ...
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Question about coloring a Cube [closed]

The vertices of a cube are numbered from $1$ to $8$. (a) What are all the elements of $S_8$ which correspond to symmetries of the cube? (b) How many ways the vertices of the cube can be coloured ...
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Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?

Consider the symmetric polynomial $$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$ How can we write $P$ as a polynomial in the variables $st,t+s$? The Fundamental theorem of symmetric polynomials implies this is ...
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How do I show two pairs of elements of $S_n$ are conjugate by the same element?

Let $\alpha, \alpha’, \beta, \beta’$ be distinct non-identity elements of $S_n$. Suppose there exists $\tau \in S_n$ such that $\alpha’ = \tau \alpha \tau^{-1}$ and $\beta’ = \tau \beta \tau^{-1}$. ...
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Symmetries of Rotation matrices

Consider the discrete symmetry $C_3$ (rotations by $120^\circ$ leave system invariant). I believe then the matrix describing this action is simply the rotation matrix (in 3d) (clockwise rotation): $$...
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Need help understanding matrix representations of the symmetric group $S_3$.

I have the following map for a representations of $S_3$: $$e \mapsto \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad (1\; 2) \mapsto \begin{pmatrix} ...
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Primitive idempotents in symmetric group rings

I am reading on a paper about the character theory of symmetric groups $S_n$. There is a claim: Set $\sigma$ be an idempotent in the ring $R(S_n)$. Then $\sigma$ is primitive iff for any $r\in R(...
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What are some nice informations about the dihedral groups,alternating groups,symmetric groups.

I am an undergraduate student and I want to know some nice informations about some special groups like the dihedral group $D_{2n}$,of regular $n$-gon , alternating group $A_n$ and symmetric group $S_n$...
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Is there any element of order $420$ in the symmetric group $S_{19}$?

Is there any element of order $420$ in the symmetric group $S_{19}$? The first thing that I checked was Lagrange's theorem. But, $420$ indeed divides $19!$, so that's no good as we could only use it ...
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Symmetric groups and surjective homomorphisms [closed]

Let, $S(6)$ be a symmetric group(of order $6$). List all groups $H$ such that there exists a surjective homomorphism $f: S(6) \rightarrow H$ Any hints or suggestions on how to approach this problem?
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Generators of the symmetric group $S_n$ [transpositions]

It is clear that $S_n$ is generated by the adjacent transpositions, i.e., $S_n$ = $\langle (1,2),(2,3),...,(n-1,n) \rangle $. So I took this idea by different way. Let me take for example $S_3$. It ...
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Example of dihedral groups with same order

I have to prove or give counter example "Is it true that if $|G_1|$ and $|G_2|$ are dihedral groups of order $|G_1|=|G_2|$ then G1≅G2 " $D_{2n}=<a,b\quad | \quad a^n=b^2=1 \quad ba=a^{-1}b>$ ...
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Combinatorics for exterior power for arbitrary Specht module

The exterior powers of the standard representation are easily seen to be the representations whose Young diagrams have only boxes in the first row or first column. But, what if I start with an ...
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Rank-4 tensor and $S_4$

I'm trying to compute the symmetrized version of a rank-4 tensor $F_{ijkl}$ associated to the following Young tableau: The Young symmetrizer for this tableau should be: $$Y=Y_VY_H=[e-(13)][e-(24)][e+(...
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1answer
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For every $H \subset \operatorname{Gal}(E / \mathbb{Q})$ find the fixed field $E^H$

Let $E$ be the splitting field of $x^3 - 2$ over $\mathbb{Q}$. I proved that $E = \mathbb{Q}(\sqrt{2},\omega)$. Where $\omega$ is a primitive root of unity. And I also know that $G := \operatorname{...
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write down all the permutations of the set {1,2,3} using two-line notation

So I was asked this on a practice set and I'm a bit confused. I know how two line notation works, but for permutations does this mean that the top row will always be (1 2 3)? This is what I think the ...
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1answer
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Number of group homomorphism from $ \mathbb{Z}_{2} * \mathbb{Z}_{2} \to D_{8}$.

How many group homomorphism are there from $ \mathbb{Z}_{2} * \mathbb{Z}_{2} \to D_{8}$?. There are $5$ elements of order $2$ in $D_{8}$ and any non-trivial element in $\mathbb{Z}_{2} * \mathbb{Z}_{...
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Decomposing into irreducible $S_n$ modules, aka Specht modules.

Let $S_{\pi}$ where $\pi$ is an integer partition of $n$, denote the Specht module corresponding to $\pi$. I am trying to decompose the set of all homogeneous polynomials in $x_1,x_2,...,x_n$ ...
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Elements of an specific order in a non cyclic non abelian group

I've got a non cyclic non abelian group G = < (1 2 3 4 5),(2 5)(3 4) > which is a subgroups of S_5 and the order of G is 10. And my question is: Is there any element of order 4 in G? What I have ...
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Let be $A_n$ the alternating group $n\geq 5$. How to prove that $\operatorname{Stab}_{A_n}(x) \cong A_{n-1}$, for all $x \in \{1, …, n\}$?

MY ATTEMPT: I have 3 properties in hand: $H\trianglelefteq G \iff Hg=gH, \; \forall g \in G \iff gHg^{-1}=H, \; \forall g \in G$; When $H=gHg^{-1}, \; \forall g \in G$ then we have $H\...
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No subgroup of order $8 $ in $S_5$ is abelian..True or False??

I may be heavily mistaken but these are my partial arguments in favour of the following statement " No subgroup of order 8 in $S_5$ is abelian" If not, let there be an abelian subgroup of order 8 in ...
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1answer
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Invariant subspace and the irreducibility of a representation.

Below is one definition for the irreducible representation: For a group $G$, a $G-$representation $V$ is said to be irreducible if there is no nontrivial proper subspace of it that is $G-$...
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Explicit formula for a homomorphism $\varphi: D_4\rightarrow \mathbb{Z}_2\times\mathbb{Z}_2$

I think I should state the original problem before I begin. Let $a = (2, 4)$, $b = (1, 2, 3, 4)\in S_4$. (Symmetric group) Let $G = \left\langle a, b\right\rangle$. Prove that $H =\{id, b^2\}$ ...
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What does “orientation” of a platonic solid really mean?

Is there any rigorous definition of "orientation" of a platonic solid? Lots of books mention that the whole group of symmetries of platonic solids consists of rotations composed with reflections, ...
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The only surjective homomorphism from $S_n$ to $\{1,-1\}$ is $\text{sign}$

I want to show that if there is a surjective homomorphism $\phi: S_n \rightarrow \{1,-1\}$ for $n\geqslant 2$, then that homomorphism is $\text{sign}$, where $$\text{sign}\ \sigma = (-1)^{\text{no. of ...
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The 5 dimensional irreducible representation of $A_5$

Hello I am trying to construct the 5 dimensional irreducible representation of $A_5$. Here is my attempt: Take the irreducible character $\phi$ of $A_4$ that has the values: $\phi(1)=1, \phi(1,2)=1,...
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the intersection of two permutation groups

Let it be G= < a , b > a group formed by two permutations from S_10: a=(1 2 3 4)(5 6 7)(8 9 10) and b=(1 3)(2 4)(5 10)(6 8)(7 9) How can I know that the intersection of < a > and < b > is ...
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1answer
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symmetric groups and cyclic groups [closed]

Are all the symmetric groups cyclic groups? I know that by definition a cyclic group is a group that is generated by a single element. But if I've got a symmetric group like for example $H=\langle(2\ ...
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1answer
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With $H = \langle(2\, 1\, 3\, 4\, 5\, 6)(7\, 8\, 10\, 9)\rangle\le S_{10}$, how do I know if $(7\, 8)H$ and $(8\, 10\, 9)H$ are equal?

I've got $H = \langle(2\, 1\, 3\, 4\, 5\, 6)(7\, 8\, 10\, 9)\rangle$ a subgroup of the symmetric group $S_{10}$. How can I know if the left cosets $(7\, 8)H$ and $(8\, 10\, 9)H$ are equal? I've ...
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Matrices that commute with Elements from the Symmetric-Group and the Hyperoctahedral Group

I am wondering whether people would have references for simple examples about: Matrices that commute with elements of the Symmetric-Group, e.g. matrix $A$ as below, and the Hyperoctahedral Group, i.e....
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symmetric group - elements with an specific order

I've got H = <(2 1 3 4 5 6)(7 8 10 9)> a subgroup of the symmetric group S_10 , and I have to calculate all the elements of H which have got order 6 If I'm not wrong, H can be written like this: H=...
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Consider a necklace that can be rotated and turned over. It is a circular necklace with 12 identical beads.

Consider a necklace that can be rotated and turned over. It is a circular necklace with 12 identical beads. Let the symmetry group be denoted by G. (i) Determine the number of rotations G contains? ...
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No normal subgroup of order $2$ in $S_4$ [closed]

The elements of order $2$ in $S_4$ are seen as six transpositions + three products of two disjoint transpositions. How do I prove that these are not normal?
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What is the asymptotics of $gf_2(S_n)$?

Suppose $G$ is a finite group. Let's define $n$-th generating fraction of $G$ as $gf_n(G) := \frac{|\{(a_1, ..., a_n) \in G^n| \langle a_1, ..., a_n \rangle = G\}|}{|G|^n}$. What is the asymptotics ...
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How to find a particular element of $S_n$ that conjugates one subgroup to another

Suppose you have two subgroups $H, K$ of $S_n$ that are given to be conjugate. How do you go about finding an element $\sigma$ of $S_n$ such that $\sigma H \sigma^{-1} = K$? For two elements $\alpha,...
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1answer
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Find cosets of $H$ in $G$

Let $G$ = $\operatorname{Sym}(\{1,2,3,4\})$ and let $ H = ⟨(1,2,3,4),(2,4)⟩$. Write out all the cosets of $H$ in $G$ So, I know that $G$ contains $4!= 24$ elements, because it's the permutation group....
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Proving/Disproving (via formal proof) that the Alternating group $A_n$ is a subgroup of the Symmetric group $S_n$

intuitively this makes sense and i can conceptualize how this would work, but i struggle to formalize a proof to express what my ideas are... Henceforth, Let n be an element of $ \mathbb Z^+ $ and ...
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How to determine the smallest $\ m\ $ with $\ G\le S_m\ $? [duplicate]

Let $\ G\ $ be a finite group with $\ |G|=n\ $ Caley's theorem states that $\ G\ $ is isomorphic to a subgroup of the symmetric group $\ S_n\ $ However, in many cases, the smallest positive integer $...
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1answer
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Orbits of pairs of multi-indices under the diagonal action of the symmetric group

This question concerns a statement made on page 168 of Dipper, R. and Donkin, S., 1991. Quantum GLn. Proceedings of the London Mathematical Society, 3(1), pp.165-211. I have tried to include all ...
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Proof that given $p$ prime and $n \in \mathbb{N}$, $n!$ divides $(p^n - 1)(p^n - p)\cdots(p^n - p^{n-1})$.

I was hoping to show this by Lagrange's theorem. As the general linear group over $\mathbb{F}_p$ (the field with $p$ elements), $\mathrm{GL}_n(\mathbb F_p)$ has order $(p^n - 1)(p^n - p)\cdots(p^n - p^...
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1answer
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Prove for $\sigma \in S_n$, $\sigma(a_1, … , a_n)\sigma^{-1}= (\sigma(a_1),…, \sigma(a_n))$ [duplicate]

How do I even begin this problem? Does it involve induction or something like induction? Would it be more helpful to instead prove it as $$\sigma(a_1,..., a_n) = (\sigma(a_1),...,\sigma(a_m))\sigma\ ?...
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Orders of Elements in Symmetric Groups

We define the symmetric group $S_n$ to be the set of all permutations of the first $n$ natural numbers. Moreover, we define the set $T_n$ as the set of all possible orders of elements in $S_n$ i.e. $...
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1answer
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Homomorphic image of an alternating group

I'm solving the following problem: If $f:S_n\rightarrow S_n$ is a group homomorphism, prove that $f(A_n)\subseteq A_n.$ (Here, $S_n$ is a symmetric group of degree $n$, and $A_n$ is an ...
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1answer
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Symmetry group of a line segment in Euclidean spaces

I am self studying the chapter of Symmetry Groups from Gallian's Abstract Algebra. There I encountered the following paragraph " It is important to realize that the symmetry group of an object ...
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General Cayley's theorem: Let $G$ be a group and $H$ a subgroup of $G$ such that $[G:H]=n$.

QUESTION: General Cayley's theorem: Let $G$ be a group and $H$ a subgroup of $G$ such that $[G:H]=n$. Then the quotient $G/H_G$ is isomorphic to a subgroup of $S_n$. ANSWER: In the answer I found, ...
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1answer
34 views

Show that $\operatorname{sign}\sigma=(-1)^{pq}$ for a particular $\sigma\in S(p+q)$

Let $S(n)$ be the symmetric group on $\{1,2,\ldots,n\}$. Let $\newcommand{\ss}{\sigma}\ss\in S(p+q)$ be the element with $$\ss(1)=p+1,\ss(2)=p+2,\ldots,\ss(q)=p+q,\ss(q+1)=1,\ss(q+2)=2,\ldots,\ss(p+q)=...
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1answer
60 views

Does $\{a,b,c,c^2\}$ generate the same group as $\{a,b,c\}$?

Is generated group by $\{a,b,c,c^2\}$ same as group generated by $\{a,b,c\}$? I think the answer is YES. But here is a paragraph of J. Wolf's Book: Let $\triangle_8$ denote the regular octahedron (...
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Prove that $Z(S_n)=\{1\}$ for all $n\geq 3$.

I'm going to type down here the answer I found in a manual solutions. However I have a question about it. And I hope I can get help here on this site. Well, I saw that there are several other issues ...
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There exists a group element $\sigma\in S_7$ under some conditions?

It is given that $\sigma\in S_7$ where $S_7$ is a symmetric group. Do there exist an element $\sigma$ such that $\sigma^{20}=\sigma$ and $\sigma\ne e$ where $e$ is an identity element? My attempt: ...

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