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

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2
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1answer
61 views

How to reach $Syl_p(H)\neq Syl_p(G)$?

I am really trapped to tis problem: If $G$ is a simple finite non-abelian group then for every proper subgroup $H$ and prime $p$ ($p||G|$) we have $Syl_p(H)\neq Syl_p(G)$. I thank for any help.
8
votes
0answers
426 views

How strong is the statement that Thompson F is amenable?

Justin Moore's proof turned out to have an error I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I ...
5
votes
3answers
706 views

Exactly one nontrivial proper subgroup

Question: Determine all the finite groups that have exactly one nontrivial proper subgroup. MY attempt is that the order of group G has to be a positive nonprime integer n which has only one divisor ...
2
votes
1answer
63 views

Bounding the index $|G' \cap H : H'|$ for $G$ a finitely generated nilpotent group

Let $G$ be an infinite finitely generated torsion-free nilpotent group. Let $H$ be a finite-index subgroup of index $n$. Let $G^\prime$, $H^\prime$ denote the derived groups of $G$ and $H$ ...
1
vote
0answers
57 views

What sort of conjugacy class data is realizable?

Let's say a cardinal-valued function $r$ of cardinal numbers (which is eventually zero) is conjugal if there exists a group $G$ in which the number of conjugacy classes of order $\alpha$ is given by ...
6
votes
1answer
449 views

Algorithm to find conjugacy classes of subgroups/elements (in matrix groups)?

I'm looking for a simple (=doable to implement by myself) algorithm to compute the conjugacy classes of elements and subgroups of a given subgroup of $\text{P}{\Gamma}\text{L}(n,q)$. So given a group ...
3
votes
1answer
111 views

Is the number of cosets in $G/H$ dependent only on size of $G$ and $H$?

This seems like something that should be obviously true, yet when my friend asked me how I knew it, I couldn't come up with a proof. And now I'm wondering if it's possible that this could somehow ...
0
votes
2answers
76 views

Proof of Existence of a Free Group (one particular step in the process)

I'm working through a lemma which is used to prove the existence of a free group on a set $S$. The setting is this: $S$ is a set, and $G$ is a group, and $f:S\to G$ is a map such that the image of ...
6
votes
3answers
453 views

Image of determinant on symplectic/orthogonal matrix group

Let $\mathbb K$ be a field, $n\geq 1$, and $G=GL_n({\mathbb K})$ be the group of invertible $n \times n$ matrices with coefficients in $\mathbb K$. For $J\in G$, we can define (in analogy to ...
3
votes
1answer
88 views

Reference for: $G$ discrete iff the measure algebra $M(G)$ is weakly amenable.

I search the reference for the proof of the following theorem: Let $G$ be a locally compact group. Then the group $G$ is discrete if and only if the measure algebra $M(G)$ is weakly amenable. The ...
4
votes
3answers
105 views

Group as a product of subgroups

Let $G$ be a finite group, and $H,K$ be proper non-trivial subgroups such that $H\cap K=1$, and $HK=G$. Is it necessary that one of these subgroups is normal in $G$?
0
votes
3answers
142 views

Normal subgroups of order 12 in $S_3\times S_3$

What are the normal subgroups of order $12$ in $S_{3} \times S_{3}$? I know that all the subgroups of order $12$ in $S_{3} \times S_{3}$ are isomorphic to the dihedral group of order $12$.
1
vote
0answers
95 views

(Symmetric) group acting on a graph

I'm currently reading "Groups acting on graphs" by Dunwoody and Dicks, and I came across section 9 (p. 39), which states that if $G$ is a group which acts on a graph $X$, then there is a group $P$ ...
1
vote
4answers
142 views

Semi-direct product of different groups make a same group?

We can prove that both of: $S_3=\mathbb Z_3\rtimes\mathbb Z_2$ and $\mathbb Z_6=\mathbb Z_3\rtimes\mathbb Z_2$ So two different groups (and not isomorphic in examples above) can be described as ...
1
vote
2answers
222 views

Commutator subgroup problem

Let $G$ be a group and let $H$ and $K$ be subgroups of $G$. The commutator subgroup $[H,K]$ is defined as the smallest subgroup containing all elements of the form $hkh^{−1}k^{−1}$, where $h \in H$ ...
0
votes
2answers
301 views

how to find the order of an element in a quotient group

Consider the quotient group $\mathbb{Q}/\mathbb{Z}$ of the additive group of rational numbers. Then how to find the order of the element $2/3 + \mathbb{Z} $ in $\mathbb{Q}/\mathbb{Z}$.
2
votes
2answers
84 views

The complement of $K$ in $G$

There is a definition which help us to understand the semi-direct product well: Let $K$ be a subgroup of a group $G$. Then a subgroup $Q\leq G$ is a complement of $K$ in $G$ if $K\cap Q=1$ and ...
0
votes
2answers
96 views

Can $a, b\in H\Rightarrow (ba)^{-1}\in H$ be a subgroup test?

Consider the following conditions: Let $H$ be a nonempty subset of group $G$. $a, b\in H\Rightarrow (ba)^{-1}\in H$. ($*$) Since $H\neq\emptyset$, there exits $a\in H$. Then one can use ($*$) ...
3
votes
2answers
221 views

Intersection of distinct maximal subgroups in a finite simple group

Suppose $G$ is a finite simple group in which every proper subgroup is abelian. If $M$ and $N$ are distinct maximal subgroups of $G$ show that $M \cap N = 1$. My plan for this problem is to use ...
7
votes
4answers
2k views

Automorphism group of the quaternion group

Let $Q_8$ be the quaternion group. How do we determine the automorphism group $Aut(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that $Aut(Q_8)$ ...
2
votes
2answers
173 views

A presentation for $\mathbb Q_8\rtimes_{\phi}\mathbb Z_3$?

We know one of the presentation of $\mathbb Q_8$ is: $$\mathbb Q_8=\langle a,b,c|ab=c,bc=a,ca=b\rangle$$ and if we want to construct the semi-direct product of $\mathbb Q_8\rtimes\mathbb Z_3$; this ...
0
votes
1answer
254 views

Simple example of a short exact sequence of groups

I am new to this and would like to understand $$0 \overset{a0}{\to} B \overset{a1}{\to} A \overset{a2}{\to} A/B \overset{a3}{\to} 0, $$ where $B \subset A$ and they are both Abelian groups. Also maybe ...
3
votes
1answer
152 views

Lattices inside matrix groups $SL_2(K)$

I am currently a second year undergraduate majoring in math and our university is offering an opportunity for undergraduates to do a project over the summer break. I have spoken to my professor who is ...
6
votes
2answers
98 views

Why is positivity of first entry sufficient for a matrix in $\mathrm{SO}(1,3)$ to be in $\mathrm{SO}^{+}(1,3)$?

This is a result physics books tell all the time, that the branch of proper Lorentz transformations with positive first entry forms the identity component of Lorentz group. In mathematical language, ...
2
votes
4answers
272 views

Does there exists an abelian or nonabelian group G where $|Z(G)|=p^2$ for p a prime?

I have known cases of abelian and nonabelian groups that have a center of order p but never a power of p. Is there at least a known case where $|Z(G)|=p^2$?
0
votes
1answer
161 views

Is the dihedral group isomorphic to its transpose?

Is there an isomorphism between the dihedral group and the group defined by the transpose of the dihedral group's Cayley table?
2
votes
2answers
250 views

Lang's Algebra: Herbrand quotient

I've looked around a lot and couldn't find much help (at least that I could understand) on this question - it is 1.45 in Lang's Algebra book: Let $G$ be a cyclic group of order $n$, generated by ...
0
votes
2answers
69 views

Elementary Question about Torsion Subgroups

Let $G$ be an abelian group which is killed by multiplication with the integer $n\geq 1$. Let $n=a\cdot b$ with $a,b \geq 1$ and relatively prime. Denote by $G[a]$ resp. $G[b]$ the $a$-resp. ...
21
votes
2answers
510 views

There are at most two prime numbers dividing $|G|$

Need just hints Let $G$ is a finite non-abelian group such that all its proper subgroups are abelian. Then there are at most two different prime numbers dividing $|G|$. I found some ideas about ...
4
votes
2answers
191 views

Show that the group $G$ is of order $12$

I am studying some exercises about semi-direct product and facing this solved one: Show that the order of group $G=\langle a,b| a^6=1,a^3=b^2,aba=b\rangle$ is $12$. Our aim is to show that ...
3
votes
3answers
125 views

Character of $S_3$

I am trying to learn about the characters of a group but I think I am missing something. Consider $S_3$. This has three elements which fix one thing, two elements which fix nothing and one element ...
1
vote
0answers
48 views

Proof of Pauli group preservation by Clifford group conjugation?!

A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words: $C(P_{1} \otimes P_{2})C^{\dagger} = P_{3} \otimes P_{4}$, with $C \in$ Clifford group and ...
0
votes
0answers
70 views

Matrix separability preservation under conjugation!?

Someone know any paper about matrix separability preservation under conjugation? A well know result is that Clifford group preserve the Pauli group under conjugation or, in other words: $C(P_{1} ...
1
vote
1answer
85 views

Virtually cyclic groups are finitely generated

A group $G$ is called virtually cyclic if it has a cyclic subgroup of finite index. Why are virtually cyclic groups finitely generated?
7
votes
5answers
873 views

Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$.

Find all subgroups of $\mathbb{Z}\times\mathbb{Z}$. I can find all the infinite subgroups of the form $n\mathbb{Z}\times m\mathbb{Z}$, where $n,m$ run over $\mathbb{Z}$. But I don't know how to ...
2
votes
1answer
267 views

Proof of the second part of Schur's lemma

Background According to PlanetMath Schur's lemma is this: Let $G$ be a finite group and let $V$ and $W$ be irreducible $G$-modules. Then, every $G$-module homomorphism $\,f: V \to W$ is either ...
3
votes
1answer
221 views

Extending a group action to a quotient group

If I have a cyclic group $G = (a)$ acting on an abelian group $A$, I need to define a natural action of $G$ on the quotient space $A/B$, where $B$ is a normal subgroup of $A$ with the property that ...
2
votes
2answers
201 views

An operation $*$ on non empty, finite set $G$ follows associative, commutative and cancellation law, prove G is abelian under $*$

Let $*$ be a binary operation defined on a non empty, finite set $G$ such that it follows associative, commutative and cancellation law.Show that $G$ under the operation $*$ is abelian. Now for $G$ to ...
8
votes
2answers
193 views

Relation between $SU(4)$ and $SO(6)$

This is more of a particle physics question than maths. Since $SO(6)$ and $SU(4)$ are isomorphic, how are the fields (say for example scalar fields of ${\mathcal{N}}=4$ Super Yang Mills in $4d$) ...
4
votes
1answer
248 views

The Bass-Serre tree for BS(1,2)

I'm trying to find the Bass-Serre tree corresponding to the Baumslag-Solitar group $BS(1,2)$. I could find only one reference from the Internet, which I don't understand, and although I have the ...
3
votes
3answers
273 views

Isomorphic subgroups, finite index, infinite index

Is it possible to have a group $G,$ which has two different, but isomorphic subgroups $H$ and $H',$ such that one is of finite index, and the other one is of infinite index? If not, why is that not ...
2
votes
1answer
118 views

Group theory question about an equivalence relation

Question: Describe the partitions of the equivalence relations for the map: $$f:(x,y) \mapsto x$$ I had a different question on my homework, but I'm not really sure what the question is asking, so ...
2
votes
1answer
255 views

Find the order of a group G from its presentation

Suppose G is a group defined by the presentation $G=\langle u, v\mid uv^2=v^3u, ~u^2v=vu^3\rangle$, is $G$ finite or infinite? If it is finite, what is its order? In general, I want to know whether ...
4
votes
1answer
199 views

When abelian group is divisible?

By definition, group $G$ is divisible if for any $g\in G$ and natural number $n$ there is $h\in G$ such that $g=h^n$. Let $A$ be abelian group with no proper subgroups of finite index. How can I prove ...
2
votes
1answer
154 views

Question about residually torsion -free nilpotent

Let $G$ be a f.g. residually torsion-free nilpotent group. Let $x$ be a nontrivial element of G, then there is a normal subgroup $N$ of $G$ such as $G/N$ torsion-free nilpotent, $x \notin N$. Let ...
0
votes
1answer
57 views

Index of Automorphism groups of graphs with colour class of bounded size.

I am trying to understand group theoretic algorithms for graph isomorphism problem. In page number 64 and 65 of this book chapter, the author has bounded the index of the concerned automorphism groups ...
2
votes
0answers
66 views

Are $\bar{G_i}(\cong G_i)$ in $\prod_{i=1}^n G_i$ unique?

We know that: Theorem: if $G=\prod_{i=1}^n G_i$ be the direct product of groups $G_1,G_2,...,G_n$ then there exists normal subgroups $\bar{G_i}\cong G_i( i=1...n)$ of $G$ such that ...
2
votes
1answer
110 views

Could a surface bundle over a circle have free fundamental group?

Specifically, I was wondering if the surface was non-compact with infinitely generated free fundamental group, could the surface bundle itself have infinitely generated free fundamental group. In this ...
4
votes
2answers
233 views

Sylow subgroup in finite groups

Let $G$ be a finite group of even order. Also in this group for every $p$ the number of Sylow $p$-subgroups is not equal to $1$. By Sylow's theorem we know that the number of Sylow $p$-subgroups in a ...
3
votes
2answers
144 views

Example of a subgroup for normality [duplicate]

Give an example of a group $G$ and a subgroup $H$ of $G$, such that for some $g\in G,\ g^{-1}Hg\subset H$ i.e. $g^{-1}Hg$ is properly contained in $H$.