# 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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### Number of involutions in $S_n$? [duplicate]

I was having some fun with the number of involutions (didn't know they were called that) in the symmetric group $\Psi(S_n$), and tried to come up with a simple formula for it, I'm like, so close to ...
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### Simple modules for the image of $\mathbb C S_\ell$ in $\mbox{End}(V^{\otimes \ell})$ and Schur-Weyl duality

Let $V = \mathbb C^n$ and let $\{v_1,\cdots,v_n\}$ be a basis over $\mathbb C$. For $\ell\geq 1$, the symmetric group $S_\ell$ acts on the $\ell$-th tensor power $V^{\otimes \ell}$ by permuting the ...
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### How does Sage order its symmetric group elements?

In Sage, the symmetric group is a list. For instance if $G$ = SymmetricGroup($3$), we have \begin{align*} G & = e \\ G & = (1,3,2)\\ G & = (1,2,3) \\ G &= (2,3)\\ G &...
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### Ways to express the identity permutation as precisely $2r$ transpositions in $S_n$

The question is as stated in the title. We know the identitiy permutation can be expressed as the product of even numbers of transpositions, but fixing the number of transpositions and asking how many ...
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### Example Calculation: Mackey's theorem for $S_5$

I'm trying to apply Mackey's theorem to a toy example where I start with the trivial $S_3\times S_2$-module $V$, induce to $S_5$ and restrict back to $S_3\times S_2$. The version of Mackey's theorem I'...
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### What is multiplicity in isotypic decompositions

I've been looking into the Quantum Schur Transform, first introduced by this paper. The explanation of the procedure goes like so: Consider the two groups: $S(N)$, the symmetric group on $N$ elements, ...
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### Conjugacy classes in a subgroup of symmetric group

Let $H=\langle (12)(34),(456)\rangle\cong S_4$ be a subgroup of $S_6$. It is known that there are five conjugacy classes in $S_4$ and the representatives are $(1), (12), (12)(34), (123), (1234)$. I ...
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### Intuition on "sharper" Cayley theorem

In "A Book of Abstract Algebra" by Pinter, chapter 16.G is the following exercise: If H is a subgroup of a group G, let X designate the set of all the left cosets of H in G. For each element ...
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### Spanning set of symmetric invariants of tensor powers

Let $M$ be a module over a commutative ring $R$ and let the symmetric group $\Sigma_n$ act on $M^n$ and $M^{\otimes n}$ by permuting factors. For $v \in M^n$, let $\text{Stab}(v)$ denote the ...
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### Subgroup Requirements

Let $H = \{g \in S_5 : g(2) = 2\text{ and }g(4)=4\}$. Is this a subgroup ? I know the identity element $\in H$ as this is $(1)(2)(3)(4)(5)$ which satisfies the requirements of the original group. For ...
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EDIT: PROOF BY INDUCTION: If there are no inversions then we have an identity permutation, so the base case holds. INDUCTIVE STEP : Lets fix some $r \in \mathbb{Z}^{+}$ and take any $\pi \in S_n$. If $... 0 votes 1 answer 102 views ### Show$\phi : S_3 \rightarrow S_3$given by$\phi(f^{i}g^{j}) = f^{2i}g^{j}$is an automorphism Show$\phi : S_3 \rightarrow S_3$given by$\phi(f^{i}g^{j}) = f^{2i}g^{j}$is an automorphism I'm not really sure how to begin. I know that I need to show that$\phi$is a group isomorphism, but the ... 5 votes 1 answer 60 views ### transitive subgroups in$S_n$that are isomorphic to$S_k$for$k\leq n$Assume that there is a transitive subgroup$H$of$S_n$w.r.t. the standard action$S_n\curvearrowright \{1,2,\cdots, n\}:=X$such that$H\cong S_k$. Is there any sharp estimate on the upper bound of ... 2 votes 0 answers 49 views ### Is there any nonabelian group of order$n$that isn't a subgroup of$S_{n-1}$? [duplicate] As far as I know, the only groups of order$n$that aren't subgroups of$S_{n-1}$are cyclic groups with prime power order and the Klein four-group. Is there any nonabelian group of order$n$that isn'... • 3,853 5 votes 1 answer 63 views ### Product of Permutation Representation Characters Consider the action of$S_n$on$x_i$, where$x_i$is a set of$i$-element subsets of$X =${$ 1, 2, ..., n$} (so$|x_i| = {n \choose i}$). Now, let$\pi_i$be the character of the permutation ... 0 votes 0 answers 36 views ### Extending the hamming distance on Sym$(X)$to linear groups I was reading of the hamming distance on the symmetry group of a finite set$X$. The distance was defined as follows: $$\text{d}^H (\sigma_1 , \sigma_2) = \frac{| { x \in X : \sigma_1(x) \neq \sigma_2(... 0 votes 0 answers 19 views ### Symmetric group characters Is there any relation between Littlewood–Richardson coefficients and the characters of the symmetric group? Can there be a relation between \chi_\lambda(\mu) and c^\lambda_{\mu \mu}. • 11 0 votes 1 answer 82 views ### Symmetric vector least squares solution I have the similar problem as the Symmetric linear least squares solution. The least square problem of mine is that I want to find$$ minimize || Ax-b ||^2,  where A\in m\times n,  x,b \... 1 vote 0 answers 13 views ### Linear involution for Specht modules Let$n$be a positive integer and$\lambda$be a partition of$n$, which we identify with its Young diagram. Let$S^{\lambda}$be the Specht module associated to$\lambda$. Here the Specht modules are ... • 1,988 1 vote 1 answer 25 views ### Show that for even integers$ n $there exists an element$ g \in D_{n} $:$ \operatorname{ord}(g)=2 $and$ \operatorname{sgn}(\varphi(g))=1 $Let$ n>3 $, let$ \Delta_{n} $be a regular$n$-corner, and let$ D_{n} $be the dihedral group which is the symmetry group of$ \Delta_{n} $. If we number the vertices of$ \Delta_{n} $, we ... 0 votes 2 answers 35 views ### Find the number of elements of$A_4$composed of the product of two non-disjoint transpositions. Find the number of elements of$A_4$composed of the product of two non-disjoint transpositions. There are following$12$elements in$A_4=\{e, (123), (132), (124), (142), (134), (143), (234), (243), ...
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I'm trying to understand a proof about the splitting criterion for the conjugacy classes of $A_n$, but I think I'm getting bogged down in earlier terminology. Here's a setup. Definitions: Let $G$ be ...