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

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4
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1answer
45 views

Problems with a proof involving graphs and groups

I'm studying an article that is the main literature when it comes to non-commuting graph : this article. Originally, a non-commuting graph of a group (denoted by $\Gamma_G$) is a graph whose vertices ...
-2
votes
4answers
57 views

Prove the only homomorphism between groups with coprime orders is trivial. [on hold]

Deduce that if $\gcd(|G|, |H|)=1$ then the only homomorphism $\phi : G \rightarrow H$ is trivial.
0
votes
1answer
30 views

Q has no maximal subgroups.

Theorem: If $R$ is a ring with 1 and $I$ is a ideal in $R$ such that $I \neq R$, then there is maximal ideal $M$ of the same kind as $I$ such that $I\subseteq M$. Note:- IF $R$ has no unity it is not ...
1
vote
2answers
12 views

Prove the order of the group homomorphism of an element divides the order of the element.

Let $\phi : G \rightarrow H$ be a group homomorphism. Prove $\forall g \in G$, the order of $\phi(g)$ divides $g$. I've gotten to the point where I've shown that if, $ord(\phi(g)) < ord(g)$ then ...
0
votes
0answers
16 views

Decomposition of an algebraic group into unipotent radical, component group and Levi subgroup

Consider the following statement : Given an algebraic group $G$ (over $\mathbb{R}$ or $\mathbb{C}$), there is a decomposition $$G = (G/G^o) \times \left( (G^o/R_u(G^o)) \ltimes R_u(G^o)\right)$$ ...
0
votes
1answer
18 views

Problem showing Newtons Laws are invariant under the Euclidean Group

I am trying to show that the equations of motions of physics are invariant under the Euclidean group $E_N$ for $N=3$. Therefore we have Newton's Laws as: $$m\frac{d^2 ...
-2
votes
1answer
33 views

Isomorphisms with factor groups

Let H and K be normal subgroups such that H$\vee$K=G and H$\cap{K}$=$\left\{{e}\right\}$, where H$\vee$K=. Prove that G/H$\simeq$K and that G/K$\simeq$H. I know elements of H$\vee$K should be of the ...
2
votes
0answers
38 views

Groups of order $8n$ have at least five distinct conjugacy classes

It was brought to my attention by Kevin Dong that every finite group whose order is a multiple of 8 must have at least five distinct conjugacy classes. This can be seen as follows: If $|G| = 8n$, ...
1
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2answers
30 views

Classifying groups of order 4

Let $G$ be a group of order $4$ then either $ G \cong C_4$ or $G \cong C_2 \times C_2$ the proof my lecture gave goes as follows: Let $x \in G$ then by Lagrange $\text{ord}(x)$ divides $|G| =4 $ so ...
1
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0answers
22 views

Prove Grigorchuk group is self similar.

Where I can find a proof of self similarity of Grigorchuk group. I read it somewhere that Grig group follows this interesting property so I read about it but could not find a proof anywhere. It was ...
1
vote
2answers
53 views

What are the three subgroups of $\mathbb{Z}_4\times\mathbb{Z}_6$ of 12 elements?

My formatting didn't work in the title, here is the question again: What are the three subgroups of $\mathbb{Z}_4\times\mathbb{Z}_6$ of 12 elements? I know that this group does not have order 24 ...
1
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2answers
35 views

Is $\mathbb{Z}\times\mathbb{Z}/((6,5),(3,4))$ is finitely generated?

Let $A$ be the quotient of the free abelian group $\mathbb{Z}^2$ by the subgroup generated by $(6,5)$ and $(3,4)$. the question is $A$ is finitely generated? and if Yes. Can we Decompose it into a ...
2
votes
0answers
24 views

Examples for Burnside problem.

What are some examples for Burnside Problem- example of an infinite finitely generated torsion group - except Grigorchuk group. I have studies Grigorchuk group as an counterexample which was first ...
5
votes
1answer
38 views

Frobenius kernel is regular normal elementary abelian p-subgroup?

I'm attempting Exercise 3.4.6 in Dixon & Mortimer's book on Permutation Groups: Let $G$ be a finite primitive permutation group with abelian point stabilisers. Show that $G$ has a regular ...
0
votes
1answer
32 views

About the special linear group $SL(n,\,\mathbb{Z})$

I found the following relation: $$SL_n(\mathbb{Z})=\langle e+e_{12},\, e_{12}+\dots+e_{n-1, n}+(-1)^{n-1}e_{n1}\rangle$$ where $e_{ij}$ is the matrix with its $(i,\, j)$th entry $1$, and all other ...
2
votes
1answer
44 views

Is GRP a subcategory of SET, or not? [duplicate]

This is the notion of a subcategory $\mathscr{D}$ of a given category $\mathscr{C}$ which I use: it consists of a subcollection of the collection of objects of $\mathscr{C}$ and a subcollection of the ...
-2
votes
0answers
29 views

problem about p-cycles [on hold]

Let a be a cycle in $S_n$ so that $a\neq(1)$ And $a^p=(1)$ with $n/2<p\leq n$ and $p$ being prime. Prove that $a$ is a p-cycle.
2
votes
1answer
11 views

Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$?

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
1
vote
1answer
36 views

About ring $( \frac{F[t]}{t^2})$

I was trying to study some properties of the group $GL_2 ( \frac{F[t]}{t^2})$ where $F$ is a field with $p$ elements. $p$ is a prime. I found that $ \frac{F[t]}{t^2}$ is a discrete Valuation ring. ...
2
votes
1answer
41 views

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$?

If $N_1, N_2, \dots, N_n$ are distinct unique Sylow subgroups, how do we know $$N_i \cap (N_1N_2 \cdots N_{i-1}N_{i+1} \cdots N_n)=\{1\}$$
2
votes
2answers
75 views

Are $(\mathbb{R} - 0, \times)$ and $(\mathbb{R}, +)$ the same group? What is its name?

I'm trying to justify to a friend why it's "not a coincidence" that $a^ba^c = a^{b+c}$, and I want to argue that it's because the structure of $\mathbb{R}$ under addition is exactly the same as the ...
1
vote
1answer
48 views

If $\left<3\right> \cong \mathbb Z_2$ and $\left<2\right> \cong \mathbb Z_3$, why isn't $\left<3\right>\left<2\right> \cong \mathbb Z_2\mathbb Z_3$

$\left|\mathbb Z_{6}\right|=6=2 \cdot 3$. Since $\mathbb Z_{6}$ is abelian, all subgroups are normal and thus its Sylow subgroups are unique. $\left<3\right>$ is the $2$-Sylow, and ...
1
vote
1answer
53 views

Annihilating Ideal of a Ring

I am stuck on how to show this. A starting hint would be helpful, and an answer (hidden) would be much appreciated. I tried supposing that there was another element in the annihilating ideal, however, ...
-3
votes
2answers
50 views

When is a group isomorphic to a proper subgroup of itself?

A infinitely generated additive group G and its subgroup K, when they are isomorphic to each other? Is there any theorem on this?
2
votes
3answers
73 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
22 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 ...
2
votes
1answer
43 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, {S1,S2,..Sn} make up the system, it can happen that, say, the intersection of S1 and S2 has order p^k, while the ...
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
100 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
65 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
46 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
144 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
121 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
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0answers
27 views

Let $G$ be abelian, $|G| = sp^i$ with $p$ and $s$ relatively prime. Show that $|K_{p^i}| = p^i$ and $|K_s| = s$ [on hold]

Let $G$ be an abelian group and $n$ be an integer. Define the map $\phi_{n}: G \to G$, $\phi_{n}(g) = g n$ since G is abelian $(hg)n = hngn$ that is $\phi_{n}$ is a homomorphism. We then have the ...
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 ...
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2answers
35 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 ...
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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 ...
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votes
0answers
25 views

generators for group for the given finite field? [on hold]

Question is "Multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a generator for this group for the given finite field" a) Z7 b)Z11 c)Z17 I don't even ...
2
votes
1answer
58 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 ...
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0answers
25 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
86 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?
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votes
3answers
44 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
65 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 ...