# 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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### Can Symmetric group on $n$ letters act non-trivially on a set with $k$ elements where $k<n$?

Let $\Sigma_n$ be the symmetric group on $n$-letters. My question is can $\Sigma_n$ act non-trivially on a set of $k$ elements say $\{1,2,\dots ,k\}$ where $k<n$? I can't think of any examples ...
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### Optimal Subgroup for Cayley's Theorem

I want to find the smallest $n$ such that $S_4 × S_8$ is isomorphic to a subgroup of $S_n$. I know that Cayley's theorem will give an $n$ but is there a way in general to find the smallest $n$? If not ...
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### Prove $Q_8$ is not contained as a subgroup in the symmetric group $S_n$, for $n<8$

Prove $Q_8$ is not contained as a subgroup in the symmetric group $S_n$, for $n<8$ I think that the key is considering the elements of order $4$ in $S_n$ but Ido not know how to face up to it. In ...
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### The alternating group An does not contain any Sn-1 isomorphic subgroup [duplicate]

It is known that we can define an action of G in the group of permutations of the set of the classes mod H, where H is a subgroup of G. The kernel of that action is the intersection of all H's ...
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### Prove there does not exist $\sigma^2=(1, 2, 3, 4)$ for $\sigma \in S_7$

I know how to prove the following using parity ie breaking down the following into its transpositions and then showing it is impossible due to the parity but was wondering if anyone had any ideas on ...
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### Image of elements of $S_n$ in the Temperley-Lieb algebra

Consider the algebra $A_n$ generated by $u_1,\ldots,u_{n-1}$ subject to relations $u_i^2=-2u_i$, $u_iu_{j}u_i=u_i$ for $|i-j|=1$ and $u_iu_j=u_ju_i$ for $|i-j| \geqslant 2$. The algebra $A_n$ is the ...
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### Constructive aspects of Dilworth's theorem for a class of finite Young's lattices

Dilworth's theorem partitions posets into the so-called chains and states that a poset of width $k$ requires only $k$ disjoint chains to decompose. It is an existential statement but constructive ...
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### Definition of transitive group

I need a definition of a transitive group that's accessible to someone who's just started learning group theory (so won't know about actions and orbits etc.). I've written the following: A ...
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### Cycle index for $S_2\times S_4$

I am trying to determine the cycle index polynomial of $S_2\times S_4$, for the purpose of finding colourings. This is what I have tried: I computed the polynomials for $S_2$ and $S_4$: Z_{S_4}(t_1,\...
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### Conjugacy class size of $S_n$

My algebra text writes for the number of cyclic structures (-,-)(-,-) in $S_4$ $\frac{{4\choose 2}\cdot {2 \choose 2}} {2}$ which should be {(1 2)(3 4), (1 3)(2 4), (1 4) (2 3)}. The binomial ...
I'd like to have a better understanding of this old valuable answer (I've tried to comment the post, but with no feedback). I'm reading (correctly?): the first two Lemmata as: $\operatorname{Aut}(S_n)... 1answer 30 views ### What is the order of$U$? Is$\{1, 7\}$an orbit from$U$? Let$G: = S_7$and$U: = \{(17), (1273)\}$. What is the order of$U$? Is$\{1, 7\}$an Orbit from$U$? Attempt: I know/see$(17)$has the order$2$and$(1273)$the order$4$, but I don't know how to ... 2answers 44 views ### Show that$G_X\le{\rm Sym}(A)$where$X\subseteq A$and$G_X=\{g\in{\rm Sym}(A)\mid g(X) = X\}.$Show that$G_X \le{\rm Sym}(A)$where$X \subseteq A$and$G_X =\{g \in{\rm Sym}(A)\mid g(X) = X\}$I'm a bit confused with this question and how to approach it. First I believe I need to show$G_X \...
I am trying to understand Collins' paper on Weingarten calculus. You can find it here: click! On page 7, between the formulas $(2.6)$ and $(2.7)$, he claims that the action \$\pi^{\otimes q} \otimes \...