A permutation group is a group consisting of permutations of some given set $M$.

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Express σ using matrix notation

Suppose σ = (3, 4, 5)(2, 4, 5) ∈ S5. (a) Express σ as the product of disjoint cycles. (b) Find the order of σ. (c) Is σ even or odd? (d) Express σ using matrix notation. (e) Find σ-1 Im not ...
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If $H<S_n$, $H$ is abelian and transitive on $\{1,2,…,n\}$, then the order of $H$ is $n$ [duplicate]

If $H<S_n$, if $H$ is Abelian and transitive on $\{1,2,...,n\}$, then the order of $H$ is $n$. So far I have: $H$ is transitive therefore the group orbit is the group itself. I know I should ...
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The structure of the Sylow p-subgroups of $Sym(n)$

For the structure of the Sylow $p$-subgroups of $Sym(n)$, there is a standard proof by using the properties of the wreath product like as in the Passman's book. But I want to understand this proof. In ...
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To show that $S_4$ has a normal subgroup of order $4$ [duplicate]

WHow to show that $S_4$ has a normal subgroup of order $4$ ? . Please help .Thanks in advance
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If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular.

If $N \trianglelefteq G$ and $G$ is $\frac{3}{2}$-fold transitive, then $N$ is transitive or semiregular. I proved that if $N$ is intransitive, then $G$ will be imprimitive and Frobenius but I don't ...
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+100

Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
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Is that true that the normalizer of $H=A \times A$ in symmetric group cannot be doubly transitive?

Is that true that the normalizer of $H=A \times A$ in the symmetric group cannot be doubly transitive where $H$ is non-regular, $A \subseteq H$ and $A$ is a regular permutation group?
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elements of order $3$ in $A_n$

Let $S_n$ be the permutation group on $n$. We know that $A_n \trianglelefteq S_n$. How many elements $ a \in A_5$ have order three. Is there any formula for finding number of elements in $ S_n $ or ...
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Primitive Permutation Group and Centralizers

Automorphism group of the Alternating Group - a proof In the above question, Derek Holt asserts that a primitive permutation groups has trivial centraliser in the symmetric group. Since I could not ...
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Every non-regular minimal normal subgroup of a doubly transitive group is primitive and simple

Prove that every non-regular minimal normal subgroup $N$, of a doubly transitive permutation group $G$, is primitive and simple. I proved that $N$ is primitive; but how I can prove that $N$ is ...
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Lagrange Theorem: left and right cosets

Thank you for taking the time to read this and help me out. This was a question I missed on my past test: "Let $G = S_3$ and $H = A_3$ ($H \subseteq G$ is a subgroup). Compute the left and right ...
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Does each element of $D4$ have an inverse in $D4$?

We are just starting the concept of permutations of objects in my class and I'm having trouble to grasp this particular question. I'm assuming it does have an inverse because of all the different ...
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Automorphism group of the Alternating Group - a proof

I was trying to read the following lemma which admit as an easy corollary the structure of the automorphism group of the alternating group on $n\geq 7$ elements. Anyway there are two points that ...
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Error in Dixon's Book?

Dixon's Book: Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive. But we have in Wielandt's book: ...
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How many 2m-permutations, consisting only of cycles of even length?

How many 2m-permutations, consisting only of cycles of even length? I have found this formula: $$Q_2(n) =((2n − 1)!!)^2$$ but how it can be proven?
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Types of ordering

Can somebody please help me understand how ordering of numbers work? There are 3 types of ordering I want to understand. Lexicographic, reverse lexicographic and Fike's ordering. How would the ...
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Problem regarding proving a permutation group

The question states: Show that the set of permutations of three objects form a group. Give the multiplication table for this group. If we take three distinct objects, the set of the ...
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A question concerning 4-cycles in $S_4$

Is it true that for all $g\in S_4$ and $f \in S_4$ a 4-cycle, then $g^{-1}fg=h$ implies $h$ is also a 4 cycle. I did a few examples, and it seems to be true, but I don't know how to prove it. Also, I ...
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1answer
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$G$ is a primitive group

Let permutation group $G$ contains a minimal normal subgroup $\neq 1$ which is transitive and Abelian. Show that $G$ is primitive. My attempts: Because of Proposition 4.4. of Wielandt's book ...
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On transitive constituents of a permuation group

Assume that the intransitive permutation group G has degree n and minimal degree n−1. If no transitive constituent of G has degree 1, then they all are faithful and all except one are regular. I ...
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If $\tau_1=(a\space b), \tau_2=(c\space d)$, why is $\tau_1\tau_2=(d\space a\space c\space )(a\space b\space d)?$

For two permutations $\tau_1=(a\space b), \tau_2=(c\space d)$, why is $\tau_1\tau_2=(d\space a\space c\space )(a\space b\space d)? \text{ (where }a,b,c,d \text{ are all distinct)}.$ I'm fairly new ...
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A question on frobenius group.

I have to do this question in my assignment... I studied frobenius group from Permutation groups by Dixon and this is the given definition in it for Frobenius group so I guess I have to assume this ...
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C*-algebra generated by the symmetric on 3 elements

I want compute $C^*(S_3)$ where $S_3$ is the symmetric group on $\{1,2,3\}$ and $C^*(S_3)$ is the (full) C*-algebra generated by $S_3$. My attempt: Since $S_3$ is a finite group, $C^*(S_3)=C_c(S_3)$ ...
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Do these permutations commute?

I read in my textbook that disjoint cycles commute. But what about $(1,2,3) = (1,3)(1,2)$ is it equal to $(1,2)(1,3) = (1,3,2)$ although they are not disjoint? And, if they are not equal how come in ...
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identity as a product of 2-cycle

In $S_n$, if $ε = α_1α_2 . . . α_r$ where $α_i$ is a 2-cycle, then r is even. I don't know how to start. Note: $ε$ is the identity of the permutation group $S_n$
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Prove that for each $\sigma \in Aut(S_n)$ $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), …, \sigma(1,n)=(a,b_n)$ [duplicate]

Prove that for each $\sigma \in Aut(S_n)$ $n \neq 6$. $\sigma(1,2)=(a,b_2),\sigma(1,3)=(a,b_3), ...., \sigma(1,n)=(a,b_n)$ for some distinct integers $a,b_1,....b_n \in \{1,...,n\}$. I was trying ...
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Permutation Matrix Proof

This is off a study guide for an exam I have in about two days. I really don't understand the problem entirely and would appreciate and help. Let $\sigma \in S_{n}$, where $\sigma$ is a permutation ...
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Generalization of a property Of $A_5$

Let $H$ and $K$ be two proper non-trivial subgroups of the alternating group $A_5$ and $\langle H,K\rangle < A_5$. We can show that there exists a maximal subgroup $M$ of $A_5$ such that ...
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Words of the Normal Form of the Presentation of a Finite Monoid

Massive Edit: After consulting with a few mathematicians at my university, I got a better understanding of what I was actually looking for. $$ \langle\ s,\ t\ \vert\ s^2 = 1,\ t^n = 1\ \rangle $$ ...
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How can I prove that every group of order 4 is Abelian. [duplicate]

Prove that every group of order 4 is Abelian. I heard the proof is just 3 lines but I don't know how to proceed. I tried proving it by showing it is isomorphic to a group of permutations, but got ...
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Is there a group which has precisely all finite groups as subgroups?

I would like to ask the following question: Does there exist a group $G$ such that every finite group can be embedded in $G$, and every proper subgroup of $G$ is finite? The closest ...
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Find a permutation with the given square or cube

Problem: find a permutation such that $x^2 = (1\;3\;4\;5\;7)$, $x\in S_7$ $x^3 = (1\;3\;4\;5\;7)$, $x\in S_7$ Must find all possible solutions for $x$. Progress I have solved for the first ...
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Permutation and symmetric Group

I have the following task: For all $i,j \in \{1,2,3,4\} $ such that $i+j=5$, let $G$ be the set of permutations $ \sigma \in S_4 $ satisfying $\sigma (i) + \sigma(j)=5$. a.) List all the ...
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How can it be proven that a cycle of length k is an even permutation if and only if k is odd?

How can it be proven that a cycle of length k is an even permutation if and only if k is odd? I know it can be done using the fact that a permutation which exchanges two elements but leaves the rest ...
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Another question on commutativity of special type of permutations

Referring to this Commutativity of special type of permutations , if $p_1p_2=p_2p_1 , p_1 \ne p_2 , p_1 \ne p_2^{-1} $ , then is it true that $U_1 \cap U_2=\phi$ ?
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Commutativity of special type of permutations

Let $p_1,p_2$ be permutations on the set $S=\{1,2,...,n \}$ and define $U_1:=\{k\in S:p_1(k)\ne k\}$ , similarly we can define $U_2$ . Now I can prove that if $U_1 \cap U_2=\phi$ , then ...
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Need help to visualise a set (reading Abelian groups)

I am reading Abstract Algebra. I cannot visualise the following example: Let $n$ be a positive integer, and consider the set $S_n$ of all permutations from the set $n = {1, 2, \ldots , n}$ to ...
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Find the subgroups of A4

I had a question I was hoping for some help on: Find all of the subgroups of $A_4$ Here is what I know: $A_4$ is the alternating group on 4 letters. That is it is the set of all even ...
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2answers
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what is the maximum order one element could have in permutation group $S_5$?

what is the maximum order one element could have in permutation group $S_5$? Tried: $|S_5| = 5!=2^3\times3\times5$ but I don't think it has anything to do with the cardinality as often seen in ...
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Logarithm of an applied permutation

Say I have a cyclic permutation $P$, a known input $x$, and a known output $y$ such that $$y = P^a x$$ for some $a$. Is there a good way to search for $a$ (i.e. better than brute force)? Are some ...
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Find this example

Let $H=\{e,(13)\}$ be a subgroup of $S_3$. Find element $a,b \in S_3$ where $bh_2ah_1 \in aH$ but $bH\ne H$. $h_1$ and $h_2$ are elements in $H$. My friend thinks that it is (123) and (132), but I ...
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What is the minimal cardinality for a generating set of the permutations?

I want to find the minimum number of permutations so that all other permutations can be obtained by multiplying the permutations of this set (taken in any quantity). In other words, I am looking for ...
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Parity of product of all permutations in $S_n$

Suppose $\alpha=$ the product of all permutations in $S_n$ for some $n$. For what $n$ is $\alpha \in A_n$, where $A_n$ denotes the set of all even permutations? Looking at $S_3$, I've determined that ...
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Automorphisms of $S(3,4,10)$

I'm trying to prove the next result: The group of automorphisms of the Steiner system $S(3,4,10)$ has a subgroup of index $2$ isomorphic to $S_6$. It'd be apprecciated if someone gave me a hand with ...
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1answer
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Mapping of elements notation - Cohn - Classic Algebra Page 13

So Cohn uses the notation that many have wanted to change to, being $xfg$ rather than $g(f(x))$, and I have had the example: Let $f,g: \mathbb{N} \to \mathbb{N}$, be given by $xf = x + 1,xg=x^2$, ...
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Solving permutation group equations

Assume we are in $S_7$. Let $\alpha^3=(1,2,3,4)$. How to solve for $\alpha$? This is what I did: $\alpha^3=(1234)$ implies that after $3$ transformations $1 \mapsto 2$. So, begin by letting $\alpha ...
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Find the centralizer in $S_7$ of $(123)(4567)$.

I am struggling to understand what the centralizer in a permutation of order $7$ means. "The centralizer consists of all elements that commute with $(123)(4567)$" but.. is there a more rigid ...
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Find number of triangle and quadrangles in $AG_2(3)$

$\textbf{Question-}$ Show that in $AG_2(3)$ there are - i) $72$ triangles ii) $54$ quadrangles iii) $4$ triangles in each quadrangle iv) $3$ quadrangles containing a given triangle My try- I know ...
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Question about the cycle type of the Frobenius Automorphism

Let $f$ be an irreducible and separable polynomial of degree $n$ over $\mathbb{F}_p$. For a finite field $F$ of characteristic $p$, we define $\phi_p:\alpha\mapsto\alpha^p$. Then we know $\phi_p$ is ...
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Determine the isometric group $G$ which transfers a square into it self

I am solving the following exercise: Determine the isometric group $G$ of the euclidean plane which transfers a square into it self. The restriction of an element $g \in G$ on the vertices of ...