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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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Point in the plane that is not fix

Let $G$ be a discrete subgroup of $M$ group of isometries of the plane whose translation group is not trivial. Prove that there is a point $p_0$ in the plane that is not fixed by any element of $G$ except the identity. Could someone give me a hint…
amir
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Number of sets mapped into a given set by permutations.

This might be a standard, simple question, and I am willing to research it but don't know where to start. I believe I know what the answer might be but can't prove it, or at least need a simple pointer. Given $S_n$, the symmetric group on…
Norman Kleinberg
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Prove in a Symmetric group, have any transposition given by product of three transpositions.

Let, there be need to show that $(i\ j)$ be given by product of three transpositions in $S_n.$ But, transpositions are composed of two elements only, so different. More restrictions on cycle size means should be able to take advantage of this rule…
jiten
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How do I compute the conjugacy classes of S_4 with their cardinality?

I asked a question earlier today if someone can help me understanding conjugacy classes. There we discussed a view aspects and now I wanted to list the conjugacy classes of $S_4$ with their cardinality, but I somehow get stuck. I have the following…
user123234
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$A_5$ is generated by all 3 cycles

There are some similiar questions, but it seems that there is some different equal definition to $A_5$. The definition I'm using for $A_5$ is that this is the kernel of the sign homomorphism- i.e all the permutations with even number of…
Ron Abramovich
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What is meaning imposing a $Z_2$ symmetry on an expression?

What is meaning concretely than an expression is $Z_2$ symmetric under a certain transformation ? Does is mean that the expression is the same when we do the transformation ? Does is mean that the expression is the opposite when we do the…
Mathieu Krisztian
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How many symmetric and antisymmetric relations are there in a set of 4 elements?

the answer to the above is 15, but I can only get 12. Say the set A = {1,2,3,4} I can only get {(1,1), ..., (1,1,1), ..., (1,1,1,1)} which gives me 12. Where is my reasoning faulty? Thank you
toyoda
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how does one decompose the defining permutation representation of the symmetric group into irreducible representations?

For all n > 2, how does one decompose the defining permutation representation of the symmetric group $S_n$ into irreducible representations?
user520403
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Determining the no. of equivalence classes in a collection of subgroups of $S_{5} $

Let $ G=S_{5} $ . Consider the set L of subgroups of G that are isomorphic to the non-cyclic group of order 4. Let us call two subgroups H and K belonging to L as equivalent if they are conjugate (that is gH$g^{-}$=K for some $g\in G$) .How many…
user-492177
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Number of conjugacy classes in permutation group $ S_n $ for some n.

I was asked to find out the number of conjugacy classes in the permutation group $ S_6 $.It is 11.But for large n it is difficult to find out .so I want to know is there any way to find out the number of conjugacy classes for large n?
suchanda adhikari
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Show that (12345) and (12) create $S_5$

Can someone tell me how I can show the following: Consider the symmetric group $S_5$ and show that $(12345)$ and $(12)$ generate the group.
lerx
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Order of the center of $Sym(n)$

With the standard notations, what's $|Z(Sym(n))|$ equal to, in terms of $n$?
user615081
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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
Suzet
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The number of product 3-cycles*2-cycles in the symmetric group $S_{5}$

Can anyone her tell me how to find the number of products 3-cycles*2-cycles such that the 3-cycles and the 2-cycles are disjoints in the symmetric group $S_5$. Thank you.
Amine El Bouzidi
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Suppose that $(S_n, \circ)$ is a symmetric group with $n \geq 3$, then a permutation of $(i,j)=(1,i)(1,j)(1,i)$

So I started studying the symmetric group, and was given this statement: Suppose that $(S_n, \circ)$ is a symmetric group with $n \geq 3$, then a permutation of $(i,j)=(1,i)(1,j)(1,i)$ And I fail to understand why is that the case. Even by…
user401855
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