# 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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### Is there a way to determine whether symmetric group $S_{n}$ has a subgroup of order $m$？

I'm interested in whether $S_{n}$ has a subgroup of order $m$. And as we know, $m | n!$ has to be hold, but it is not a sufficient condition. I have researched some other cases. If $m \le n$, then ...
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### Let $H=\{\sigma\in S_n\mid 1\le\sigma(i) \le k\;\forall i=1,2,\cdots,k\}$. Prove that $H$ is a subgroup of $S_n$ and $H\cong S_k\times S_{n-k}$.

I have trouble completing this proof. Let $k,n \in \mathbb{Z}, 0 < k < n.$ Let $H=\{\sigma \in S_n\mid 1 \leq \sigma(i) \leq k\; \forall i =1,2,\cdots,k\}.$ Prove that $H$ is a subgroup of $S_n$...
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### General theory for cyclic modules of the group algebra of the symmetric group

Let $\mathbb{C}[\mathfrak{S}_n]$ be the group algebra of the symmetric group. An element of this algebra is of the form $$v = \displaystyle \sum_{g \in \mathfrak{S}_n} a_g g,$$ where $a_g \in C$. ...
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### What is the full symmetry group of a tile in the shape of a regular n–gon?

I am trying to answer the question on exercise 4.5.6 from the book "Algebra: Abstract and concrete" by Goodman. The chapter is on the symmetries of polyhedra and in this exercise he asks me ...
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### Number of conjugacy classes in a set of subgroups of $S_5$ which are isomorphic to klein's 4 group.

Let, K be a set of subgroups of S5 (symmetric group of 5 elements) that are isomorphic to the non-cyclic group of order 4. How many conjugacy classes are there in K? I know that a non-cyclic group of ...
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### Can we choose a set to make sure the action of a permutation group transitive?

Let a finite group $G$ of order $n$ be given, so $G$ is isomorphic to a permutation group embedded in $S_n$. Can we always find a set $\Omega$ such that $G$ acts transitively on $\Omega$? (For example,...
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### Explicit isomorphism of algebras $\mathbb C[S_3]\cong \mathbb C \times \mathbb C \times M_2(\mathbb C)$

Let $\mathbb C[S_3]$ be the group algebra of $S_3$. We have an isomorphism of $\mathbb C$-algebras $$\mathbb C[S_3]\cong \mathbb C \times \mathbb C \times M_2(\mathbb C).$$ The existence of such an ...
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### An identity about the sum of the reciprocal of the irreducible character of a symmetric group evauated at identity

I just learnt character theory and I am reading the paper https://academic.oup.com/jlms/article-abstract/66/3/623/811347 which is a quite beautiful paper. Section 4.3, on p. 631 contains an inequality ...
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### Character table of S4

I am trying to understand the character table of $S_4$. I have obtained the trivial, signature and standard representations. The fourth one is the product of signature and standard. Now for the last ...
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### When is the Cayley embedding for infinite groups optimal?

By Cayley's theorem, any group $G$ naturally injects into the symmetric group $\mathrm{Sym}(G)$ of its underlying set via the Cayley embedding. Let's say this embedding is optimal for $G$, if there is ...
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### Triple-Transitivity/"Specify three know all" property of exotic transitive $S_5\subset S_6$
Let the exotic transitive subgroup $S_5\subset S_6$ act on $\{1,2,\dots,6\}$. For $1\leq i,j\leq 6$, define subsets: $$X_{ji}:=\{\sigma\in S_5\,\mid \sigma(j)=i\}.$$ Does the following properties hold ...