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

learn more… | top users | synonyms

0
votes
0answers
24 views

Permutation group with two orbits of equal size contains a permutation with no cycles length 1 [on hold]

For G a permutation group on a set X with exactly two orbits of the same size, how can we prove that there exists a permutation g in G that does not contain any cycles of length one?
1
vote
3answers
38 views

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 ...
1
vote
1answer
46 views

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 ...
0
votes
1answer
45 views

$G$ 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 ...
0
votes
0answers
26 views

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 ...
0
votes
0answers
26 views

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 ...
2
votes
0answers
21 views

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 ...
4
votes
1answer
25 views

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, $C^*(S_3)=C_c(S_3)$ ...
1
vote
2answers
42 views

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 ...
0
votes
0answers
35 views

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 $S_n$
0
votes
0answers
37 views

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 ...
1
vote
1answer
40 views

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 ...
2
votes
1answer
94 views

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 ...
0
votes
1answer
21 views

Minimal expressions of generators of a symmetric group.

${S}_n$ is the symmetric group of $n$ symbols. Let $G_n = \{\ s_n,\ t_n,\ id\ \}$ be the set of generators of ${S}_n$ where $$s_n = (1\ n)$$ $$t_n = (1\ 2\ \dots\ n)$$ $$id\text{ = identity}$$ A ...
0
votes
0answers
15 views

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 ...
3
votes
0answers
30 views

How can i prove that an element of order 5 is a 5 - cycle in S7 group? [closed]

Please prove that an element of order 5 is a 5 - cycle in any S n group. I am absolutely lost.
12
votes
0answers
202 views

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 ...
0
votes
1answer
30 views

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 ...
0
votes
0answers
45 views

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 ...
0
votes
2answers
19 views

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 ...
0
votes
1answer
17 views

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 $U_1 \cap U_2=\phi$ ?
2
votes
2answers
43 views

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 ...
6
votes
2answers
108 views

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 ...
0
votes
2answers
72 views

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 ...
2
votes
2answers
32 views

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 ...
1
vote
0answers
16 views

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 ...
1
vote
1answer
60 views

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 ...
1
vote
1answer
33 views

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 ...
1
vote
2answers
28 views

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 ...
0
votes
0answers
26 views

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 ...
0
votes
1answer
21 views

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 $xf = x + 1,xg=x^2$, ...
1
vote
1answer
47 views

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 ...
2
votes
1answer
57 views

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 ...
0
votes
0answers
15 views

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 ...
1
vote
1answer
36 views

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 ...
1
vote
1answer
16 views

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 ...
1
vote
1answer
27 views

$5|\#\text{Gal}(f/\mathbb{Q})\subset S_5 \implies \text{Gal}(f/\mathbb{Q})$ contains a $5$-cycle?

Context: Consider $$ f(x):=x^5-4x+2\in\mathbb{Q}[x]. $$ By Eisenstein's criterion, $f$ is irreducible over $\mathbb{Q}$. Since $\mathbb{Q}$ has characteristic $0$, we know every irreducible polynomial ...
2
votes
1answer
48 views

How to find out if two cycles are conjugate?

Let for example $a=(14395)(26)(78)$ and $b = (154)(2368)(79)$ be elements of $S_9$. I know that by definition, conjugate elements of a group $G$ are elements $x,y \in G$ such that $x=aya^{-1}$ for ...
1
vote
3answers
40 views

$V_4\triangleleft S_4$

Let $V_4:=\{(1\,2)(3\,4),(1\,3)(2\,4),(1\,4)(2\,3),\iota\} \leq S_4$. It is possible to show $V_4\triangleleft S_4$ by considering conjugation. However, after long thought on the matter, I don't ...
3
votes
1answer
70 views

Show that the order of $G$ cannot be divisible by $7$.

Show that the order of a proper primitive group of degree $19$ cannot be divisible by $7$. It means as following- This means that a group $G$ which acts on a set $\Omega$ of cardinality $19$ (this ...
0
votes
0answers
25 views

Permutation combination question

Say that i have 4 variables: $X_1, X_2, X_3$ and $X_4$. and $Y$ is a function of these four variables which uses four operational symbols $(+,-,*,/)$. This could be: $X_1+X_2+X_3+X_4 $ ...
0
votes
2answers
55 views

Expressing permutations as products of transpositions and identifying parity

$(156)(234)= (16)(15)(24)(23)$ Even $(17254)(1423)(154632)= (12)(13)(16)(14)(15)(13)(12)(14)(14)(15)(12)(17)$ Even How is parity determined? Is it simply even when the # of transpositions is even ...
1
vote
1answer
56 views

An iff condition for $2$-transitive groups

$\textbf{Theorem}$ - A group $G$ acts doubly transitively on a set $X$ iff $1/|G|\sum_{g\in G}|fix(g)|^2=2$. I Have no idea how to begin. If it had been finite group and finite set, then at the ...
1
vote
1answer
25 views

Stabilizer of a doubly transitive is maximal?

Is it true that if $G$ is a group acting $2$-transitively on a set $X$ , then if $x\in X$, then $G_x$ (stabilizer) is maximal in $G$. I think it must be true as a conclusion of $2$ theorems, as ...
0
votes
3answers
45 views

Subgroups of $S_4$ generated by cycles

I am new with abstract algebra and I trying to find all the subgroups of $S_{4}$ generated by the cycles : a) $(13)$ and $(1234)$ b) all cycles of length $3$ I am not sure how to start so I would ...
1
vote
1answer
37 views

Equation for a systematic permutation

A $6$ digit number is set whereby every digit can be repeated without any constraints. So one can have a number between $000001$ and $999999$. (Zeros on the left are counted). The problem: Generate ...
1
vote
0answers
76 views

Is this problem still open or solved?

Problem- Let $n\geq$ and let $T$ be the set of all permutations in $S_n$ of the form $t_k=\prod_{1\leq i\leq k/2}(i,k-i)$ for $k=2,3,4.....(n+1)$. Then find the least integer $f_n$ such that ...
1
vote
1answer
49 views

Basis of vector space invariant under group action (of symmetric group)

Suppose I have a finite-dimensional real vector space $X$ and a finite group $G$ that acts faithfully on X. The task is to find a $G$-invariant basis of $X$. This means the set of basis vectors is ...
4
votes
2answers
74 views

Is every group a permutation group?

I just read about permutation groups. Before going further this question came up in my mind. Isn't every group a permutation group? The definition says, "one-to-one mappings of a set onto itself is ...
1
vote
0answers
73 views

Primitive subgroups with different order

I have been studying the permutation group $S_{n}$ in combination with transitivity and blocks. Which leads to primitive groups. Now I was wondering if for $n=6$, so $S_{6}$ could have two primitive ...