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 $|i-j|>1$
  • $(\sigma_i\sigma_{i+1})^3 = 1$
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Why are two permutations conjugate iff they have the same cycle structure?

I have heard that two permutations are conjugate if they have the same cyclic structure. Is there an intuitive way to understand why this is?
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Can someone explain Cayley's Theorem step by step?

This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. Can someone explain this proof step by step?…
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$A_4$ has no subgroup of order $6$?

Can a kind algebraist offer an improvement to this sketch of a proof? Show that $A_4$ has no subgroup of order 6. Note, $|A_4|= 4!/2 =12$. Suppose $A_4>H, |H|=6$. Then $|A_4/H| = [A_4:H]=2$. So $H \vartriangleleft A_4$ so consider the…
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$A_n$ is the only subgroup of $S_n$ of index $2$.

How to prove that the only subgroup of the symmetric group $S_n$ of order $n!/2$ is $A_n$? Why isn't there other possibility? Thanks :)
ShinyaSakai
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Enumerating all subgroups of the symmetric group

Is there an efficient way to enumerate the unique subgroups of the symmetric group? Naïvely, for the symmetric group $S_n$ of order $\left | S_n \right | = n!$, there are $2^{n!}$ subsets of the group members that could potentially form a subgroup.…
Hooked
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elliptic functions on the 17 wallpaper groups

In doubly periodic functions as tessellations (other than parallelograms), we learned about the Dixonian elliptic functions. There are 17 wallpaper groups -- are there elliptic function analogues for the other 15 cases (not covered immediately by…
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Normal subgroups of $S_4$

Can anyone tell me how to find all normal subgroups of the symmetric group $S_4$? In particular are $H=\{e,(1 2)(3 4)\}$ and $K=\{e,(1 2)(3 4), (1 3)(2 4),(1 4)(2 3)\}$ normal subgroups?
Marso
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Find the center of the symmetry group $S_n$.

Find the center of the symmetry group $S_n$. Attempt: By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\ g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is always the trivial permutation. Suppose $a$ is in…
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Probability that two randomly chosen permutations will generate $S_n$.

Every undergraduate learns a fact about the symmetric group that $(1\,2)$ and $(1\,2\,\cdots\,n)$ generate $S_n$. Also, it is interesting to know that for a prime $p$, any 2-cycle and any $p$-cycle generates $S_p$, but an arbitrary $2$-cycle and…
Beginner
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Can a figure be divided into 2 and 3 but not 6 equal parts?

Is there a two dimensional shape (living in a plane) that can be divided into $2$ and $3$ but not $6$ equal parts of same size and shape? This question is a simpler take on this puzzling.SE question. If such a shape exists, the $3$ parts can't be…
Eod J.
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Rotman's exercise 2.8 "$S_n$ cannot be imbedded in $A_{n+1}$"

This question is about the (in)famous Rotman's exercise 2.8 in "An Introduction to the Theory of Groups." I've searched and found similar questions here and in MO, but none of them contains a valid proof. (Does $S_n$ belong as a subgroup to…
Riccardo
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Proof that no permutation can be expressed both as the product of an even number of transpositions and as a product of an odd number of transpositions

I am aware that there are a couple of well-known proofs of this theorem, but I'm specifically grappling with the proof given in Fraleigh's A First Course in Abstract Algebra (Theorem 9.15 in the textbook). Let $s$ be a permutation in the symmetric…
ryang
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$(12)$ and $(123\dots n)$ are generators of $S_n$

Show that $S_n$ is generated by the set $ \{ (12),(123\dots n) \} $. I think I can see why this is true. My general plan is (1) to show that by applying various combinations of these two cycles you can get each transposition, and then (2) to show…
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How to enumerate subgroups of each order of $S_4$ by hand

I would like to count subgroups of each order (2, 3, 4, 6, 8, 12) of $S_4$, and, hopefully, convince others that I counted them correctly. In order to do this by hand in the term exam, I need a clever way to do this because there can be as many…
Pteromys
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Normal subgroups of infinite symmetric group

I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the set $\mathbb{N}$) has exactly 4 normal…
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