Mathematics Stack Exchange
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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.

The symmetric group $S_n$ is a group consisting of all permutations of a set of $n$ elements with composition as the binary operation. You could equivalently think of it as the group of all bijective functions from a set $\{1,2,\dotsc,n\}$ to itself. The symmetric group can be generated by the functions that swap adjacent pairs of elements $\{1,2,\dotsc,n\}$. This leads the a common presentation of the symmetric groups with generators $\langle \sigma_1, \sigma_2, \dotsc, \sigma_{n-1}\rangle$ and relations

  • $\sigma_i^2 = 1$
  • $\sigma_i\sigma_j = \sigma_j\sigma_i$ for $|i-j|>1$
  • $(\sigma_i\sigma_{i+1})^3 = 1$
2772 questions
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Conjugate permutations in $S_n$ and / or $A_n$

Given two permutations , I'm asked to answer is they are conjugate permutations . The two permutations are : $ \alpha=(12)(345)(78)$, $\beta=(162)(35)(89)$. Definition: Two permutations $ \sigma,\sigma'\in S_n$ are conjugate if exists $\tau \in S_n…
JAN
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Numer of permutations that are product of exactly two disjoint cycles

Let $a_n$ denote the number of those permutations $\sigma$ on $\{1,2,3....,n\}$ such that $\sigma$ is a product of exactly two disjoint cycles. Then $a_5 = 50$ $a_4 = 14$ $a_5 = 40$ $a_4 = 11$ I tried specifically for $a_5$ and $a_4$ with a…
Hirakjyoti Das
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are identities symmetric on both sides of an equation

if the LHS of an identity is symmetric does it mean the RHS must also be symmetric? In addition how do you test if an identity in three variables is symmetric e.g let the three variables be x,y and z do you replace x with y, y with z and z with x -…
zebra1729
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A general way to determine a decomposition of the identity of a symmetric group $S_n$.

I am interested in determining a decomposition of the identity of a symmetric group $S_n$ like $$\sigma_1\sigma_2...\sigma_m=1$$ for distinct $\sigma_i\in S_n$ (distinct to remove the case $\sigma_i=1\forall i$). Some questions I would like…
levitopher
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Permutations conjugated

Show that the permutations: $\alpha= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 2 & 5 & 3 & 6 & 1 & 4 \\ \end{pmatrix} $ and $\beta= \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \\ 5 & 3 & 4 & 2 & 1 & 6 \\ \end{pmatrix} $ Are conjugated in $S_6$ and you…
Peter G
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Name of transitive group of polynomial with cubic degrees

What is the transitive group details of a polynomial where only the third power terms occur? That is $x^{3n} + a_{n-1} x^{3(n-1)} + ... + a_1 x^3 + a_0$. I need the basic theorems that state or prove this group structure. I think that the answer…
menkelh
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Orders of elements for rotational symmetries of cube

I am having lots of trouble doing this problem because I have particularly poor visualization skills. (Or maybe haven't developed them well yet). I would appreciate any help on this math problem. Here is the question: Suppose a cube is oriented…
arabhi manachra
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Determining decomposition into orbits

Decompose the set $\mathcal{M}_2(\mathbb{C})$ of $2 \times 2$ complex matrices into orbits for the following operations of $GL_2(\mathbb{C})$: left multiplication. Could someone give me a complete answer to this question? I have another "similar"…
amir
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Symmetry groups in algebra

Recently I was going over online notes regarding symmetry groups and I came across the following notation: $S_3=\{1,x,x^2,y,xy,x^2y\}$ is generated by $\{x,y\}$. What does this mean? Aren't the elements in $S_3$ of the form…
Rutherford Mark
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Combining Operation for $S_4$

If $\sigma=(1\ 2\ 3\ 4)$, $\kappa=(1\ 2)$ for $S_4$ and I want to compute $(\sigma\kappa)^2$, does it become $\sigma^2\kappa^2 = \sigma^2$ (since $\kappa^2 = 1$), which is just $(1\ 3)(2\ 4)$?
m1koto
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Transposition on an arbitrary permutation of Symmetric group may coalesce disjoint cycles?

Say, an arbitrary permutation in $S_7$ as: $$\sigma = \begin{pmatrix} 1 2 3 4 5 6 7 \\3 2 5 7 6 1 4 \end{pmatrix} \implies (1356)(47)$$ Here, $\sigma^{-1}(3) = 1,\sigma^{-1}(4) = 7, \sigma(3) = 5, \sigma(4) = 7$. To this we applied a…
jiten
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Prove that every permutation in $S_n$ may be written in terms of $(1,2)$ and one non-trivial element.

Need which property of algebra, or otherwise; to prove it.
jiten
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Confusion about size of conjugacy class formula in $S_n$

I was reading this answer The size of a conjugacy class in the symmetric group but I didn't understand point 1). Why does this contribute a factor of $2!\times2!$? After all, the 4 cycles can be moved around freely: the cycle type $(3,3,2,2)$ is an…
QuartelQuartz
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Calculator/Tool to find subgroup of $S_n$ generated by $\langle g_1 \dots g_k \rangle$

does anyone know an online utility tool that computes the subgroup of the symmetric group generated by generators, for example, I want the program to tell me $$ \langle (13)(24), (13) \rangle = \{ (1), (13)(24), (13), (24) \} $$ I’ve tried…
user651267
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generators of alternating groups?

Let $A_{5}$ be the alternating subgroup of the symmetric group $S_{5}$. Prove that $A_{5}$ is generated by the two elements $\{a=(123),b=(12345)\}$, or equivalently can we write the element $(234)$ as a composition of the two elements $a$ and $b$.
user27759
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