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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show that $x^3 = (1234)$ in $S_7$ has three solutions (and find them?) [duplicate]

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| = ...
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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 ...
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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 ...
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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) ...
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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,...
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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 $S_5,A_5,\{e\}$...
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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 ...
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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 ...
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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=...
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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) ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $H$ ...
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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 $p\in S_n$ ...
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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)...
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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+...
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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 &...
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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 $(n-1)!$
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$\sigma \in S_n$ 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 $\...
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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 ...
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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} \...
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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 ...
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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 ...
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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)...
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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 ...
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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-...
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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 ...
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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 ...
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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,...
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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 \...
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-2 votes
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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 ...
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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 ...
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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 ...
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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 ...
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4 votes
1 answer
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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), \...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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, $...
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1 vote
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Prolongation of group actions are smooth

I am working on Olver's book Applications of Lie Groups to Differential Equations and I have been trying to prove how a prolongation of a smooth action $(g,x,u) \mapsto g \cdot (x,u)= (\Xi_{g}(x,u),\...
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4 votes
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Finding the number of the group homomorphisms $G\to S_4$ ($|G|=6$) by group actions.

A homomorphism from a group $G$ of order $6$ to $S_4$ is equivalent to an action of $G$ on the set $X=\{1,2,3,4\}$. By the orbit-stabilizer theorem, every orbit must have size either $1$, or $2$, or $...
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1 answer
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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 ...
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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, $...
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