The study of symmetry: groups, subgroups, homomorphisms, and group actions.

learn more… | top users | synonyms (2)

2
votes
3answers
77 views

Why isn't $\mathbb Z_9 \cong \mathbb Z_3 \times \mathbb Z_3$ by the fundamental theorem of finite abelian groups?

I was reading the answer to this question: Explicit descriptions of groups of order 45 and the accepted answer says the Sylow $3$-subgroup is either isomorphic to $\mathbb Z_9$ or $\mathbb Z_3 \times ...
-1
votes
0answers
27 views

Group $G$ acts on set $Ω$. $|G| = 30$, $Ω$ has size 3.

Five elements of $G$ fix every element of $Ω$. What subgroup sizes are guaranteed? Without the fixed elements, by Sylow E and Cauchy theorems, there should be $1,2,3,5,6,10,15$. Are all of the ...
1
vote
1answer
23 views

Proving relations between kernels and images of a Group G

Let $G$ be an abelian group and n be an integer. Define the map $\phi_n\colon G \to G$, $\phi_n(g) = g^n$ since $G$ is abelian $(hg)^n = h^ng^n$ that is $\phi_n$ is a homomorphism. We then have the ...
0
votes
2answers
29 views

$|G| = 1155$, $N \lhd G$, $|N| = 55$, $K \leq G$, $|K|=35$. $|<N,K>|$ and $|N \cap K|$?

since $gcd(|G:K|,|N|) \neq 1$, I can't use $NK=K$ and $N \cap K = N$. I tried using Sylow $p$-subgroups, but they don't seem to help this problem. Does $NK \lhd G$ have to be true? Also are ...
3
votes
1answer
56 views

Various Intersections of Sylow p-subgroups.

I was told yesterday that in a system of Sylow $p$-subgroups of a finite group $G$, if, $\{S_1,S_2, \cdots, S_n\}$ make up the system, it can happen that, say, the intersection of $S_1$ and $S_2$ has ...
1
vote
1answer
20 views

Consider $\phi: G \to S_4\times D_{15}$ a homomorphism and onto

Q1) Prove that $G$ contains an element of order 20 Q2) Assume $\exists H\subset G$ s.t $H$ normal in $G$ and |$\phi(H)$|=60. Prove that $G$ contains a normal subgroup $K$ such that |$G/K$|=36 For ...
5
votes
2answers
104 views

Algorithm design for enumerating pairs of noncommuting elements up to conjugacy

I am trying to write some Magma code that, given a group $G$, returns a list of pairs $(x,y)$ with $x,y\in G$ such that $[x,y]\neq 1$ and such that every pair $(z,w)$ in the group with $[z,w]\neq 1$ ...
1
vote
1answer
23 views

Free product of infinite many groups is not finitely generated?

Let $\{G_i\mid i\in I\}$ an infinite family of (not trivial) groups. Is it true that the free product $\ast_{i\in I} G_i$ is not finitely generated? I think it's true, I just need confirmation.
4
votes
2answers
66 views

Elements of order three in $GL_3(2)$

How do I go about finding elements of order 3 in $GL_3(2)$? I'm currently trying to show that the automorphism group of a Klein 4-group induced by conjugation in $GL_3(2)$ is isomorphic to $S_3$ so am ...
1
vote
2answers
47 views

Question about relationships between images and kernels of a group

I have been having a reoccuring problem in my abstract algebra class with my professor defining notation and using things different from the books for homework and it's giving me difficulty to follow. ...
0
votes
0answers
24 views

Prove that any non-abelian group of order $10$ is isomorphic to $D_5$ [duplicate]

Show that any non-abelian group of order $10$ is isomorphic to $D_5 (=\{\tau^i, \sigma\tau^j:0\le i,j\le 4\}$ with $\tau\sigma=\sigma\tau^4$ and $\sigma^2=\tau^5=e).$ I want to use Sylow's theorems ...
4
votes
2answers
146 views

What is a “natural group action”?

Eg. The symmetric group on S acts on S in a natural way, for all sets S. Thanks in advance!
1
vote
1answer
39 views

Exercise in group action blocks

I am reading the book "Permutation Groups" by Dixon and Mortimer in which they discuss blocks and primitivity of group actions. An important theorem which I just read its proof states: Let $G$ act ...
2
votes
1answer
123 views

problem on permutations

In $S_{10}$, can someone explain why there is no permutation $a$ such that $a(1,2,3)a^{-1} = (1,3)(5,7,8)$?
0
votes
0answers
19 views

Prove that there exits an automorphism from $G$ to $G$ when dim G=infinite

Suppose $G$ is a vector space over $\mathbb Z_2$ . The problem is to prove that there exits an automorphism from $G$ to $G$ Now $G$ has a basis say $\{b_1,b_2,...,b_n\}$.Then any $g\in G$ can be ...
1
vote
2answers
37 views

Prove that $G$ is a vector space over $\mathbb Z_2$

Suppose $G$ is an abelian group such that all non-identity elements in $G$ has order $2$. Prove that $G$ is a vector space over $\mathbb Z_2$ Since $G$ is an abelian group only thing to show is to ...
1
vote
0answers
19 views

Is trace of regular representation in Lie group a delta function? [duplicate]

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
2
votes
1answer
62 views

Is trace of regular representation in Lie group a delta function?

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
0
votes
0answers
8 views

Commutation Relation - Generators of Semisimple Lie Group

The following is stated in a book I picked up, Group Theory for High-Energy Physicists - M. Saleem, M. Rafique. Consider an $r$-parameter semisimple Lie group of rank $\ell$. It has a set of ...
0
votes
0answers
26 views

Geometric understanding of why $D_1\cong \mathbb{Z}_2$

From what I understand, $P_2$ is the line joining $(-1,0)$ to $(1,0)$ and then $P_1$ is the point $(1,0)$. This is due to defining $P_n$ (the regular polygon with # of points $n$) to be formed by the ...
2
votes
2answers
87 views

Conjugacy classes of the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$

Consider the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$. This group does not appear to be easy to work with! Does anyone know what this group is called? I am trying to find its ...
0
votes
2answers
47 views

What would the notation G/H mean in terms of groups and subgroups?

What would G/H mean in terms of subgroups? Would it most likely mean The compliment group of H in G?
0
votes
3answers
45 views

An example of a group that that has an order of M that is abelian?

Theory: If G is a finite abelian group, p is prime and p divides the order of G then G has an element of order p. Can anyone think of a counter example for a number n that is not prime, divides the ...
2
votes
0answers
67 views

If a certain group action fixes every element $x$ such that $x^4=1$, then the action is trivial

This is a question from chapter $4D$ of Isaacs' Finite Group Theory. Let $A$ act via automorphisms on $G$, where $G$ is a $2$-group and $A$ has odd order. Show that if $A$ fixes every element $x$ in ...
0
votes
1answer
49 views

Using generators to write representations

Let $G=D_{12}=\{a,b\mid a^6=b^2=1, bab=a^{-1}\}$. Also let $A=\begin{pmatrix} e^{i\pi/3} & 0 \\ 0 & e^{-i\pi/3} \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. ...
2
votes
1answer
46 views

Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order.

$G$ is a finite group. $\rho$ is a representation of $G$, then $\rho \mapsto \rho(g)$ is a $1$-dimensional representation. Show that if $\det \rho(g)=-1$ for some $g \in G$ then $G$ has a ...
0
votes
1answer
38 views

Only one cancellation law? Then $G$ may not be a group…

Suppose that the following result is known: "Let $G$ be a finite set, closed with respect to an associative product and that both of the cancellation laws are valid. Then $G$ is a group with ...
2
votes
1answer
24 views

Prove that the inner product is the number of orbits of $G$ on $X \times Y$.

Let $X,Y$ be $G$-sets and $\mathbb{C}[X], \mathbb{C}[Y]$ the corresponding permutation representations. Prove that the inner product is the number of orbits of $G$ on $X \times Y$. Ive tried: ...
0
votes
5answers
34 views

Product of subgroups which is not a group - infinite case

It is well known that a product of subgroups might not be a subgroup. I give an example here. Would you have an example of an infinite group having two infinite subgroups whose product is not a ...
2
votes
1answer
56 views

Number Theory : Is a complete residue system modulo $n$ a group?

I was working my way through some basic number theory problems, when in the chapter on "Introduction to Group Theory," I came across the following: Show that for every positive integer $n$, the ...
1
vote
0answers
32 views

Is there a general method to calculate the generators of the subgroups of $\textrm{GL}(n,F)$?

I know this might be a very bad/broad question, but after going through a few practice problems for finding linearly independent generators for some of the easier subgroups of $\textrm{GL}(n,F)$ ...
1
vote
0answers
33 views

Direct limit of quotient groups

For a subgroup $H$ of $G$, we denote $\langle H\rangle $ be the smallest normal subgroup of $G$ containing $H$. (That is, the normal closure of $H$ in $G$.) Suppose that ...
2
votes
1answer
29 views

What is the process behind finding a Cayley permutation representation.

For example, let's find the Cayley permutation representation of $\mathcal D_3$ in $S_6$. $\mathcal D_3 = \left<r,s \mid r^3=s^2=1, rs=sr^{-1}\right>$. Write, \begin{pmatrix} 1 & 2 & ...
4
votes
3answers
92 views

Let $G$ be a finite simple group. Suppose that $A, B < G$, $G = AB$ and $A$ is an Abelian group. Is it true that $A \cap B=1 $?

Let $G$ be a finite simple group. Suppose that $A$ and $B$ are proper subgroups of $G$, $ G = AB$ and $A$ is an Abelian group. Is it true that $A \cap B=1 $ ? I checked it with some examples and it ...
2
votes
2answers
130 views

How to show that the union of an infinite sequence of subgroups is a subgroup?

I'm self-studying from Algebra: Chapter 0 by Aluffi. I was working on Exercise 6.6, and I can do the first part (which I've included for context), but I'm having trouble with the second part. Here is ...
0
votes
2answers
59 views

How do I find the permutation with the highest order in a symmetric group?

My professor gives this text, but I don't understand what it's saying, could someone explain it to me? Let $M(n)$ denote the largest order of an element in $S_n$. By Theorem 1 $M(n)$ is the largest ...
1
vote
2answers
50 views

I have a question about group G which satisfies Inn(G) char Aut(G) and Z(G)={1}.

Let $G$ be a group which satisfies $Z(G)=\{1\}$ and $Inn(G) \space \mathbb{char} \space Aut(G)$; then every automorphism of $A=Aut(G)$ is an inner automorphism. ($H \space \mathbb{char} \space G$ ...
1
vote
2answers
41 views

For $n \ge 3$, every subgroup of $A_n$ with index $n$, is isomorphic to $A_{n-1}$

For $n \ge 3$, every subgroup of $A_n$ with index $n$, is isomorphic to $A_{n-1}$. Any idea to solving it?
0
votes
1answer
31 views

generating system of the kernel of a module-transformation

Let $G ≠ 1$ be a group and A a commutative ring. Now, the group ring $A[G]$ is naturally an A-module. Next, let's consider the transformation: $$\phi: A[G] \to A, \sum_{g \in G} a_gg \mapsto \sum_{g ...
1
vote
2answers
52 views

Show that $\rho$ must be 2-dimensional

Let $G=D_8=\langle g,h |g^4=h^2=1, hgh=g^{-1} \rangle$. One can show that $G$ has $4$ $1$-dimensional representations. From first principles (no character theory). Suppose $\rho$ is an ...
1
vote
2answers
56 views

How to read a cycle graph?

As an important tool for visualizing some small finite groups it is useful to know how read such graph, and with time trying to make sketch of them by my own. I would like to know, for a start, how ...
2
votes
1answer
85 views

When is $G\cong\operatorname{End}(G)$?

$\newcommand\End{\operatorname{End}}$Let $G$ be an Abelian group. Are there sufficient conditions for the existence of an isomorphism $G\cong\End(G)$, where $\End(G)$ is considered a group under ...
2
votes
1answer
59 views

Need of proof of associative property of symmetric group

I'm a novice in learning group theory.Recently I'm learning about symmetric group. I understood the closure property,identity property and inverse property of symmetric group but I don't understand ...
0
votes
1answer
30 views

How it works as cyclic group, somebody do explain it.

For a geometrical realisation of a cyclic group, let $S$ be the circle, in the plane, of radius $1$ , and let $\rho_n$ be a rotation through an angle of $\frac{2 \pi}{n}$. Then $\rho_n \in A(S)$ and ...
1
vote
1answer
63 views

Why the orbit is of dimension $12$?

Let $SL_3$ acts on the variety consisting of all nilpotent $3$ by $3$ matrices over $\mathbb{C}$ by conjugation. Let $S_p$ be the orbit of the matrix $$ a=\left( \begin{matrix} 0 & 1 & 0 \\ 0 ...
0
votes
2answers
45 views

If $F$ is a free group with basis $X$ and $N = \langle \{ g^2 : g \in F\} \rangle$, why is $F/N$ elementary abelian of rank $|X|$?

This seems to be a standard trick - take the subgroup $N$ generated by all squares of elements in a group $G$. Then $N$ is normal, since the conjugate of a square is a square, and $G/N$ is abelian ...
3
votes
1answer
75 views

Let $G$ be a finite group and let $P$ be a Sylow p-subgroup with $N_G(P)=P$. Prove that $P$ is not contained in any proper normal subgroup of $G$.

Let $G$ be a finite group and let $P$ be a Sylow p-subgroup with $N_G(P)=P$. Prove that $P$ is not contained in any proper normal subgroup of $G$. We can suppose $P \subseteq K\lhd G$. Since $P$ is ...
3
votes
2answers
75 views

Why do we think of group compositions as multiplication?

This has bothered me for some time: The composition in a group is usually denoted $xy$ or $x\cdot y$. Powers (note the word) are denoted by $x^n$, inverses by $x^{-1}$, and the neutral element by $1$. ...
1
vote
1answer
68 views

$G$ is a finite group,with eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ s.t. index $[G:N]$ divisible by 56, not by 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. I will start by ...
1
vote
1answer
24 views

Are free products of finite cyclic groups perfect?

I read that $\text{PSL}(2,\mathbb{Z})$ is isomorphic to $\mathbb{Z}_2*\mathbb{Z}_3$, which is a perfect group. Then, in general, for natural numbers $n$ and $m$, when is $\mathbb{Z}_n*\mathbb{Z}_m$ ...