# 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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### Finitary Alternating Groups

If $X$ is an infinite set, then the finitary alternating group on $X$ can be defined in the following equivalent ways: the group of all even permutations on $X$ under composition the kernel of the ...
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### Prove that a normal subgroup $G$ of $S_4$ with $(12)\in S_4$ is equivalent to the entire group $S_4$ [duplicate]

Consider a normal subgroup $G$ of $S_4$. The simple transposition $(12)\in G$. Prove that $G=S_4$. I have already proved that the above case implies that $(12),(23),(34)\in G$. These are all the ...
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As title. This is question 52 in Chapter 5 of Gallian’s Abstract Algebra, 10th edition. My current line of logic is as follows. We have $x^3 = (1234)$, which gives us $|x^3| = 4$, implying that $|x| = ... 10 votes 1 answer 128 views ### Permutation Groups: Find$x$such that$x^5 = (12345)$I am wondering about how to solve question 35 from chapter 5 (Permutation Groups) from the 10th edition of Gallian’s Abstract Algebra. The full question is as follows: What is the smallest$n$for ... 0 votes 1 answer 20 views ### Sources that representations on$S_n$are real-valued I've been told that representations of$S_n$are real-valued and that there are some sources for that. However, I can't find anything, does anybody have a book / paper that proves this? Thanks in ... 2 votes 1 answer 54 views ### Proving there aren't any more subgroups for$S_3$I'm currently doing an exercise to find all the subgroups of$S_3$, with a hint given that there are exactly$6$and then to prove that no more subgroups exist. Take $$() \equiv e, (12) \equiv x, (13) ... 1 vote 1 answer 18 views ### The commutativity of (12)(34) and (12) sufficient to say that : the S_5-conjugacy class of (12)(34) is also an A_5-conjugacy class? I read in a document about the symmetric group. I came across a paragraph that I didn't understand why, it's the following: One checks that (123) commutes with the odd permutation (45). Therefore,... 1 vote 0 answers 60 views ### Prove that there is no homomorphism from S_5 onto a group of order 24. Prove that there is no homomorphism from S_5 onto a group of order 24. My solution: Let G be a group such that |G|=24. Denote \phi: S_5\to G. The normal subgroups of S_5 are S_5,A_5,\{e\}... 4 votes 1 answer 51 views ### Understanding von Dyck's theorem I'm trying to understand how to use Von Dyck's theorem to prove that S_3 \cong D_3. I believe I have a correct sketch, but I'm very fuzzy on the details, mainly because I haven't seen free groups ... 0 votes 0 answers 18 views ### Signature of a permutation couple I was working on a problem when I found out I needed to know the signature of a permutation of the form : \begin{equation} (\sigma_A,\sigma_B) \end{equation} meaning that (\sigma_A,\sigma_B) is ... -3 votes 1 answer 29 views ### Why is S_n=⟨(12),...,(1n) ⟩? [closed] I'd like to know why S_n=⟨(12),...,(1n) ⟩ (the permutation group), I have a proof but I didn't get the last step, taking just that we have: \sigma=(a_1 \ a_2)=(1 \ a_1)(1 \ a_2)(1 \ a_1) where n=... 2 votes 0 answers 67 views ### Number of inversions in S_n I have a problem about some result appearing just before the proposition 1.5.2 in Combinatorics of Coxeter Groups by Bjorner, Brenti. It's about the inversions in S_n. I don't understand why 1.26) ... 0 votes 0 answers 71 views ### Proof idea: S_3 \cong D_3 [duplicate] I am trying to show that S_3 is isomorphic to D_3 as groups. The definitions I'm working with are$$ S_3 = \left \langle (12), (123) \right \rangle, \; D_3 = \left \langle r, s \mid r^3 = s^2 = 1, ... -2 votes 0 answers 78 views ### Unsure about how to orbit-counting theorem works [closed] The symmetric group S3 acts on a set X consisting of 84 elements. Given that each transposition fixes 20 elements and each three-cycle fixes 54 elements, compute the number of orbits. This question ... 5 votes 1 answer 173 views ### Is$S_2 \wr S_k$contained in$(S_a \times S_b) \wr S_{k-1}$? I'm working on my thesis and I want to prove a theorem but I need the following to be true:$S_2 \wr S_k$is not isomorphic to a subgroup of$(S_a \times S_b) \wr S_{k-1}$where$a,b < 2k$. Does ... 1 vote 0 answers 11 views ### What transitive action of a permutation group connects the Prufer 2-group? Let$X$be the dyadic rationals in the half-open unit interval. The graph$G$over$X$having the vertices$(x,x+2^{\nu_2(x)-1})$and$(x,x-2^{\nu_2(x)-1})$is connected. It's just the infinite ... 3 votes 1 answer 64 views ### Representation of$\mathfrak{S}_p$over$\mathbb{F}_p$I'm learning modular representation theory from the 3rd part of Serre's book. Through the process called "reduction mod.$\mathfrak{m}$", we obtain representations in positive characteristic ... 1 vote 1 answer 69 views ### No Subgroup of$S_7$of order 15 If$S_7$is the symmetric group on 7 elements and$H \leq S_7$is a subgroup with$|H| = 15$, then$H$is a subgroup of order$pq$where$p \nmid q-1$for$p=3$and$q=5$. This would imply that$H$... 0 votes 1 answer 34 views ### About a proof for$S_n$being generated by adjacent transpositions I know this is a standard result and I'm looking at the proof here (Theorem 2.0.1) I have one quick question about this proof. The proof uses induction on$n$by considering any permutation$p\in S_n$... 0 votes 0 answers 8 views ### asymptotic of a maximum defined over the symmetric group Let$d,k$be positive integers. I am looking for an asymptotic (when$n\to+\infty $) of the quantity $$\max_{\sigma\in\mathfrak S_k}\left\{\sum_{i=n+1}^{n+k}2^{\sigma(i-n-1)}\left[\frac{i-\sigma(i-n-1)... 2 votes 1 answer 44 views ### Abelian group of order pq is a subgroup of S_{p+q} Let G be a finite abelian group of order pq, where p<q are both primes. I want to show that G is isomorphic to a subgroup of S_{p+q} (but is not isomorphic to any subgroup of order S_{p+... 0 votes 0 answers 16 views ### Normal subgroups chains and symmetries of geometric shapes There is a beautiful parallel between the normal subgroups chain of symmetric groups and the symmetries of 2D/3D shapes. Here's the tables for each symmetric group. S2: rotation of 2 vertices of an &... 0 votes 1 answer 85 views ### Order of largest subgroup of S_n [closed] I know that S_{n-1} is a maximal subgroup of S_n, but is it also maximum? I.e., what's the size of the largest subgroup of S_n? Is it (n-1)! 2 votes 0 answers 51 views ### \sigma \in S_n does not commute with any odd permutation if and only if the cycle type of a consists of distinct odd integers. This is (part 1 of) Exercise 4.3.21 in D&F 3ed Abstract Algebra, which I have to prove: Show that \sigma \in S_n does not commute with any odd permutation if and only if the cycle type of \... 5 votes 1 answer 61 views ### Homomorphisms from S_3 to \mathbb{Z}/10\mathbb{Z} I want to check if my line of thought is correct. We need to find all homomorphisms \phi: G=S_3\rightarrow H=\mathbb{Z}/10\mathbb{Z}. We already know that \phi(g)=\bar{0} for all g\in G is a ... 0 votes 0 answers 52 views ### Cayley's theorem - How many cycles of each type exists Suppose we have a group isomorphism G \cong H \leq S_{q+1} . Let \chi_{q+1} denote the number of (q+1) - cycles \in H and suppose q is an odd prime, then I've shown that \frac{q-1}{2} \... 4 votes 1 answer 171 views ### Symmetric Group of an amorphous set An amorphous set is an infinite set, which is not the disjoint union of two infinite sets. The existence of such sets is consistent with ZF. I am wondering, if there are any interesting remarks to be ... -5 votes 2 answers 84 views ### Yes/No: There always exists an injective homomorphism from G into S_n. [closed] Let G be a finite group of order n\ge2. Is the following statements true/false? There always exists an injective homomorphism from G into S_n. My attempt: I found the answer here. I think ... 1 vote 0 answers 42 views ### Let G be a non abelian simple group. Show that {\rm Aut}(G^n) is isomorphic to {\rm Aut}(G) \wr{\rm Sym}(n). I got this question but don't know how to answer it. Let G be a non abelian simple group. Show that {\rm Aut}(G^n) is isomorphic to {\rm Aut}(G) \wr{\rm Sym}(n). I already know that {\rm Aut}(G)... -1 votes 1 answer 78 views ### Prove that there exists finitely many groups of order n. Every finite group is isomorphic to some permutation group. Any group of order n can be embedded into S_n. (We say that group G_1 is embedded into G_2 if there is f:G_1\to G_2 that is ... 0 votes 0 answers 37 views ### Show S_3 is the semi-direct product of A_3 by H=\{(1),(12)\} Show S_3 is the semi-direct product of A_3 by H=\{(1),(12)\} How would you prove such a question? Showing H \cap N = \{id\} is fine, where N=A_3, but how would I show that A_3H=G for semi-... 2 votes 0 answers 43 views ### Do bitwise rotations and increments modulo 2^n generate the symmetric group? Let's say we have two operations we can perform on a binary number of length n: Right-rotation, where the most significant bit is taken off and inserted in the one's place, pushing all other digits ... 2 votes 2 answers 64 views ### If G has no subgroup of index 2 and G\leq S_n, then G \leq A_n. I am currently reading Abstract Algebra by Dummit & Foote. Discussing some techniques about the Sylow theorems they prove the proposition mentioned in the title (Proposition 12 (1), p. 204, 3rd ... 0 votes 1 answer 39 views ### Action on Quotient group Let G =\langle A \rangle \leq S_n be a permutation group. The natural action of G on \Omega , where |\Omega|=n is defined as follows: f:G \times \Omega \rightarrow \Omega such that f(\sigma,... 0 votes 1 answer 58 views ### Why does this Galois Group contain an (n-1)-cycle? I am looking at this Theorem and proof from : https://www.researchgate.net/publication/320835842_The_Inverse_Galois_Problem_4th_year_project (page 12) For n > 3, there exists a polynomial f \in \... -2 votes 1 answer 65 views ### Understanding homomorphism and non-homomorphism from A_4 to S_6 Could anyone help me how to find an example of a function f: A_4\to S_6 that is not a homomorphism and one f that is a homomorphism? And is A_4\times S_6 isomorphic to S_6\times A_4? Thanks so ... 3 votes 1 answer 36 views ### Given n, do we have a formula for the greatest Hook number of an n-box Young diagram? Obviously the least Hook number is 1, by considering boxes stacked vertically or horizontally. Is there a formula for the greatest possible Hook number ? EDIT: The Hook number of a Young diagram is ... 1 vote 2 answers 68 views ### How to construct an injective homomorphism from GL(2,\mathbb{Z}_2) into S_4? I am struggling to devise an injective homomorphism from GL(2, \mathbb{Z}_2) to S_4, in particular construction which actually allows me to verify whether its a homomorphism or not. Eg, we could ... 0 votes 0 answers 27 views ### Is the conjugacy class of an element in the alternating group equal to that in the symmetric group? I tried looking it up online but I have found nothing, therefore I am unsure whether what I am doing is right, I'd like to also know what I am doing wrong in the proof. Remember that if two ... 4 votes 1 answer 201 views ### Trying to understand a proof of Laplace expansion I am trying to understand the ProofWiki proof of Laplace expansion for determinants. I understand the first equation$$ D = \sum_{\sigma} {\rm sgn} (\rho) {\rm sgn} (\sigma) \prod_{j=1}^n a_{\rho(j), \... 0 votes 0 answers 46 views ### permutation representation of the symmetric group$𝑆_𝑛$and its trace I have this algebra task which I have encountered problems with proving a specific identity for, Consider the permutation representation of the symmetric group$𝑆_𝑛$, which gives a group ... 0 votes 1 answer 74 views ### Subgroups of the symmetry group Question: Let$A = \{\{x,y\}:x,y\in\{1,2,...,n\},x\neq y\}$. We want to show that$\exists G\leq \text{Sym(A)}$s.t.$G\cong S_n$and$\{g(\{1,2\}):g\in G\}=A$Attempt: For the first part I'm pretty ... 0 votes 1 answer 53 views ### Is there an easy way to find the derived group of$S_5$? In order to find the derived group of$S_5$I've tried using Lagrange’s Theorem to find the order of the possible subgroups but$O(S_5)=2^3\cdot 3 \cdot 5$so there are too many possible subgroups to ... 1 vote 1 answer 36 views ### exterior powers of Tensor Product of vector spaces Let$k$be a field of characteristic zero and$V_1, V_2$two finite dimensional$k$-vector spaces. I would like to show that$$\wedge ^2 (V_1 \otimes V_2) = (\operatorname{Sym} ^2(V_1) \otimes \... 1 vote 2 answers 105 views ### Prove that${\rm Inn}(S_n)$isomorphic to$S_n.$Show that${\rm Inn}(S_n)$isomorphic to$S_n$for$n\ge3$. To do this, if I define some isomorphic function say$\phi$, where$\phi: S_n \to{\rm Inn}(S_n)$, then show that$\phi$is bijective (by ... 1 vote 0 answers 77 views ### Does there exist a group$G$such that${\rm Aut}(G) = S_6$? I thought this would be an interesting question, since every other symmetric group on$n$elements is possible as an automorphism group, since${\rm Aut}(S_n) \cong S_n$for$n \neq 2, 6$. Obviously,$... 1 vote
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### Homomorphisms from $C_2\times C_3$ to $S_4$
How many homomorphisms are there from $C_2\times C_3$ to $S_4$ are there? (Using kernel and image to describe). My thoughts/attempt: Determine homomorphisms by the image of the domain's generators. We ...
### For α and β ∈ Sn, define α ∼ β if there exists a σ ∈ $S_n$ such that $σασ^{−1}$ = β. Show that ∼ is an equivalence relation on $S_n$
My attempt is below. Could I please get feedback on it. I am not so sure that it is correct. Let α,β,σ ∈$S_n$. Since $S_n$ is a group, we know that it contains an identity. Let e be the identity. So, \$...