Should be used with the (group-theory) tag. Symmetric group is a group consisting of all permutations of given finite set with composition as the binary operation.

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Rotations of a cube

I am trying to create a program using Python 3 which must simulate the rotations of a cube. However, I am struggling to figure out how to rotate that cube. I have the following formulas: ...
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26 views

commutator (derived) subgroup of S3

how can i calculate it easily? i showed that the commutator group of S3 is generated by (123) in S3 using the fact that S3 is isomorphic to D6 and relation in D6 but that was tedious...are there any ...
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1answer
42 views

When the elements of maximum order are $n$-cycles in $S_n$?

If the elements of maximum order in $S_n$ are $n$-cycles, then we can guess with few computations that $n$ must be at most $4$. How can we prove this? I tried the case in $S_{2n+1}$, the symmetric ...
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1answer
22 views

Dimension of unordered configuration space

I'm working on $X = \mathbb{R}^n$ and I'm considering the set of unordered sequence of points of $X$. Considering $F(p) = \lbrace (x_1, \dotsc, x_p) \in X^p ; i \neq j \implies x_i \neq x_j \rbrace$ ...
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If $|H|=112$ then $A_7\cap H \lhd H$?

I posted this because Alex Clark asked in chat and I'm not sure how to proceed. Let $G$ be a group such that it has a fixed subgroup isomorphic to $A_7$, which we denote simply by $A_7$. Let $H$ be a ...
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33 views

Finding all homomorphisms [duplicate]

How can I solve this question Find all group homomorphisms from $\mathbb{Z_{5}}$ into $\mathbb{Z_{12}^{x}}$?
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40 views

How to find the number of solution of $x^n=1$ in the group $S_n$?

Suppose that $S_n$, the symmetric group of order $n!$ is given and for given $m\in \mathbb N$ fixed, we are to find the number of solutions to $\theta^m=e, \theta\in S_n$. Can someone tell me or give ...
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60 views

“Every element of Sym$(n)$ has order at most $n$”

I was doing mini-test involving a True/False section and came across the following statement. Every element of $Sym(n)$ has order at most $n$ I admit I had gotten this incorrect as I had thought ...
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Is there a good way to break down the order of the centraliser in a symmetric group?

I recently rediscovered the rather nice formula for the order of the centraliser of a permutation in the symmetric group and its realtionship with conjugacy classes. I wondered whether we could say ...
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Is there a synthetic definition of the $0$-Hecke monoid of $S_n$?

Background. Let $n$ be a nonnegative integer, and let $S_n$ denote the $n$-th symmetric group. The $0$-Hecke monoid $H_0\left(S_n\right)$ is defined to be the monoid given by generators $t_1, t_2, ...
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73 views

Cayley's theorem

As according to Cayley's theorem "Every group is isomorphic to a subgroup of some symmetric group". Now my question is: the additive group of real numbers is isomorphic to which permutation group... ...
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101 views

Space of arbitrary rotations of a cube

Suppose I have a cube $[-1,1]^3\subset\mathbb{R}^3$. I am allowed to rotate it about any angle/axis through the origin rather than just $90^\circ$ about the coordinate axes, e.g., by applying ...
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2answers
59 views

Ordered pairs of permutations in symmetric group

How many ordered pairs $\left(\alpha_1,\alpha_2\right)$ of permutations in symmetric group $S_n$ that commute: $$\alpha _1 \circ \alpha _2 = \alpha _2 \circ \alpha _1\,,$$ where $\alpha _1, \alpha _2 ...
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1answer
34 views

Can $S_n$ be embedded in $GL_{n-1}(\mathbb Z)$ ? [closed]

How to prove that $S_n$ is isomorphic with a subgroup of $GL_{n-1}(\mathbb Z)$ ?
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1answer
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Two group elements induce the same permutation on $A$ if and only if they are in the same coset of the kernel.

Page 113 - Dummit and Foote - Group actions Two group elements induce the same permutation on $A$ if and only if they are in the same coset of the kernel. What does this mean? Two ...
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Why does $\sum_{\sigma\in S_n}q^{\ell(\sigma)}=\frac{(1-q)(1-q^2)\cdots(1-q^n)}{(1-q)^n}$?

This is a known result, but I can't find a proof. Why does $$ \sum_{\sigma\in S_n}q^{\ell(\sigma)}=\frac{(1-q)(1-q^2)\cdots(1-q^n)}{(1-q)^n}? $$ Here $\ell(\sigma)$ is the length of $\sigma$, ...
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If $g=(n-j,\dots,n)$ and $\sigma\in S_{n-1}$, why are the inversions of $g\sigma$ the union of the inversions of $\sigma$ and $g$?

I can't see why a claim I'm reading is true. If $\sigma\in S_n$, let $R(\sigma)=\{(i,j):i<j,\ \sigma(i)>\sigma(j)\}$, i.e., $R(\sigma)$ is the set of inversions of $\sigma$. The set ...
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Is there any neat way to show $\phi$ is a homomorphism?

In Michael Artin's Algebra (chapter 2, page 50, example 2.5.13) the author illustrates a homomorphism from $S_4$ (all permutations of indices $(1,2,3,4)$) to $S_3$ (all permutations of indices ...
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1answer
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Norm $\Vert \cdot \Vert$ on the symmetric group $S_n$

If we define a real valued function $\Vert \cdot \Vert$ on the $n^{th}$ order symmetric group $S_n$ satisfying following conditions $$\begin{align} & \|x\|=0\iff x=\omega\,\,\,(\text{identity ...
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2answers
51 views

How to solve conjugation equations in group theory

Given the permutations $(12)(34)$ and $(56)(13)$ find $a$ such that $$a^{-1}xa = y$$ I just realized that I don't know how to solve this exercise. My book don't even give examples of how to solve ...
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1answer
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Counting number of bijections

The question is: Let $S = \{a,b,c,d\}$ and let $X : = \{f\colon S \to S \mid f \, \, \text{is bijective and } f(x) \ne x \, \, \text{for each}\, \, x \in S \}$. What is $|X|$? Is there a simple ...
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If $G = S_5$ and $H = \{g \in G \mid g^{5} = e\}$ how could I determine and prove whether or not $H$ is a subgroup of $G$?

I think that the this group contains the 5 element cycles and the identity e but overall I'm not sure how to prove that the product of the 2 members of H is also a 5 cycle or e.
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1answer
39 views

Presentation of the symmetric group of 5 symbols.

I am trying to write the presentation of the symmetric group $S_{5}$. We know that $S_{5}$ is generated by $a=(1,2)$ and $b=(1,2,3,4,5)$. Using this I am trying to write presentation of $S_{5}$. My ...
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Possible orders of the elements of the alternating group $A_n$

What are the possible orders of the elements of the alternating group $A_n$, that $n\in \{P,P+1,P+2,2P,2P+1\}$, where $P$ is the set of primes numbers?
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1answer
34 views

Representation-theoretical reasons for positivity of product of two Schubert polynomials?

In the Wikipedia article on Schubert polynomials there is a claim that there are representation-theoretical reasons for the product of two Schubert polynomials to have nonnegative coefficients when ...
3
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1answer
31 views

Which permutation am I? Or: what is a bijection $f:S_n \rightarrow \{1,2,\ldots,n!\}$ such that we can compute $f(\beta)$ easily?

Let $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$ and assume that $S_n$ is ordered in some way, i.e., $$S_n=\{\alpha_1,\alpha_2,\ldots,\alpha_{n!}\}.$$ We are able to choose this ordering on ...
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113 views

More Symmetric than the symmetric groups?

So I was considering the following question. Is there a group of size n! (or less), that contains a larger number of subgroups than $$S_n$$? In thinking about this problem one obvious contender that ...
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Subgroups of $S_n$ with exactly one fixed point for each element all have the same fixed point.

Let $G$ be a subgroup of $S_n$ (where $n$ is a positive integer) such that each non identity element $g\in G$ has exactly one fixed point. Prove there is an element of $[n]$ that is fixed by every ...
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proving Caley's Theorm using the conjugacy operation instead of left multiplication

I need to prove Caley's theorem that each group G is isomorphic to a sub group of S(n). Wherever I check it is proven using the operation of multiplying from left side. that means $f_g(x) = g*x$ I ...
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Set of generators for $A_n$, the alternating group.

The problem is this: Prove that $A_n = \langle (123),(124),\ldots,(12n)\rangle$. I had cogitated this problem for quite awhile, and haven't been able to come up with anything. The only good idea ...
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1answer
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There is no injective morphism from $\mathbb S_7$ to $\mathbb A_8$

I am trying to show that it doesn't exist an injective group morphism $f:\mathbb S_7 \to \mathbb A_8$. If there is an injective morphism from $\mathbb S_7$to $\mathbb A_8$ then $\mathbb S_7$ is ...
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2answers
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Prove or disprove $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$

I need to prove or disprove: To the group $A_5$ has a subgroup that isomorphic to $\mathbb{Z}_6$ My attempt: I just wrote all the details that I know: element in $A_5$ should be in form like ...
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Maximum order of an element in Alternating group of degree 10.

What is the maximum order of any element in $A_{10}$? My attempt: I tried this problem. But I am not sure about the answer. My answer is 21 because 10 can be written as 7+3. $A_{10}$ can have ...
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1answer
36 views

What does it mean to find elements in $S_9$ that are “not cycles”?

I came across this wording in the following question. Some clarification on what this means and how to approach this problem would be helpful. Thanks!
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Help with some simpler symmetric group $S_n$ problems.

I apologize if the problems seem trivial but I have not been able to find example problems or solutions to some of these questions. Could someone please confirm my attempts are correct or not? ...
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1answer
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How to use class equation for determining the center of $S_4$

How to use class equation for determining the center of $S_4$ $$|G|=|Z(G)|+\sum_x [G:C_G(x)]$$ So I guess I need to find $$|G|-\sum_x [G:C_G(x)]=|Z(G)|$$ Well $|S_4|=4!=24$ and $C_G(x)$ is the set ...
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1answer
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transitive subgroups of the symmetric group on $2(d+1)$ elements: can I always do at least $d$ permutations?

I have a set of $2(d+1)$ elements which are labelled as pairs $\{e_i, a_i\}_{i=1}^{d+1}$, transforming under some transitive subgroup of the symmetric group. This can be thought of as a regular ...
3
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1answer
41 views

On the number of conjugacy classes in $S_n$.

Let $\sigma=(1,2,11)(3,4)(5,6,7,8,9)\in S_{12}$. I am willing to get the number of conjugates of $\sigma$. Clearly if $\tau$ be one such, then it must have the same cycle type. So in other words, we ...
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The Fifteen Puzzle and $S_n$ [duplicate]

I was studying permutation groups from the book "Abstract Algebra and Applications" by Karlheinz Spindler in which page 553 I came across the following interesting problem. It is on the famous "The ...
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1answer
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Symmetric Polynomials and Automorphisms of Complex Polynomial Rings

I asked a version of this question earlier, but it was very imprecise and poorly formatted, so I decided to create a new question. Suppose we have an ordered set of $n(n-1)/2$ distinct polynomials ...
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1answer
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Product of $n(n-1)/2$ polynomials of the same degree is symmetric

I am trying to prove a simple fact about polynomials in the multivariate polynomial ring $\mathbb{C}[x_1,x_2,...x_n]$, for $n \gt 3$ but I've been getting stuck. EDIT: After a comment by Tad I ...
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1answer
51 views

$SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3)) \cong \mathbb A_4$

I am trying to prove that the quotient group $SL_2(\mathbb Z_3)/Z(SL_2(\mathbb Z_3))$ is isomorphic to $\mathbb A_4$. I could show that $SL_2(\mathbb Z_3)$ has $24$ elements (one can see this ...
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Composition of groups

Let's say we have a system of interacting particles that can divided into two populations. The symmetry group of each population is $G$, and the two populations are identical, so that I can exchange ...
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90 views

How to prove that a given group is isomorphic to Sym(4)?

Given a specific group with 24 elements, I want to prove that it is isomorphic to Sym(4). To begin with, I calculate the orders of my group's elements and they come out as in the order statistics for ...
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1answer
18 views

Need help understanding the precise meaning of “unique factorisation of disjoint cycles”

Below is taken from my linear algebra course lecture notes: Some facts about permutations of $\{1,2,\dots,n\}$: Every permutation is a product of disjoint cycles which commute. For example ...
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Cycle Structure of a Permutation Based on the Binary Representation

Define a permutation $\sigma$ on the set $X=\{1,2,...,n\}$, $n$ is a natural number as follows. Given a non-negative integer $k$, let $s(k)=\frac{b+1}{2}$, where $b=\max\limits_c\big(c2^k\le n, ...
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Representing a 5-cycle as a product of transpositions

Dr. Pinter's "A Book of Abstract Algebra" shows that: $$(12345)$$ can be written as the following product of transpositions: $$(54)(53)(52)(51)$$ How can the first representation, $(12345)$, be ...
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30 views

Permutation module $M^\lambda$ as induced module

If we let $r$ be a natural number, $\lambda$ be a partition of $r$, $\Sigma_r$ be the symmetric group on $r$ numbers, we can define the following $K\left[ \Sigma_r \right]$-module: $M^\lambda := ...
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Conjugation of permutations

In the group $S_n$ I usually use the fact that if $(a_1 a_2 \dots a_r) \in S_n$ is an r-cycle and $\sigma \in S_n$ then $\sigma (a_1 a_2 \dots a_r)\sigma^{-1} = (\sigma(a_1)\sigma(a_2) \dots ...