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

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10
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2answers
3k views

Painting the faces of a cube with distinct colours

I don't think this is solved by Burnside's Lemma since there is a condition that each side is painted a different colour. The question is as follows. If I had a cube and six colours, and painted ...
3
votes
1answer
100 views

Subgroup relations in $GL(3,\mathbb Z)$

There are 73 conjugacy classes of finite subgroups in $\operatorname{GL}(3,\mathbb{Z})$. If you take 73 representatives, you will find group-subgroup relations between them. There must exist an ...
7
votes
1answer
198 views

Maximal Subgroups and order of a group

I encountered the following exercise in Isaacs' Algebra: "Suppose a group $G$ has only one maximal subgroup. Prove that the order of $G$ must be a power of a prime". I think I've proven this for ...
0
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2answers
2k views

Prove that every element of a finite group has an order

I was reading Nielsen and Chuang's "Quantum Computation and Quantum Information" and in the appendices was a group theory refresher. In there, I found this question: Exercise A2.1 Prove that for ...
2
votes
1answer
109 views

Every abelian $p$-group is an image of some direct sum of cyclic $p$-groups

Every abelian $p$-group is an image of some direct sum of cyclic $p$-groups. This is an exercise on page 113 of Derek J.S. Robinson's A course in the Theory of Groups. I don't know why it is ...
32
votes
1answer
965 views

Automorphisms inducing automorphisms of quotient groups

Let $G$ be a group, with $N$ characteristic in $G$. As $N$ is characteristic, every automorphism of $G$ induces an automorphism of $G/N$. Thus, $\operatorname{Aut}(G)\rightarrow ...
12
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5answers
2k views

How can you show there are only 2 nonabelian groups of order 8?

It's often said that there are only two nonabelian groups of order 8 up to isomorphism, one is the quaternion group, the other given by the relations $a^4=1$, $b^2=1$ and $bab^{-1}=a^3$. I've never ...
1
vote
1answer
266 views

Reference: Group Theory

I want to read a book on group theory "Endliche Gruppen I" by Huppert, Blackburn etc. I do not know, whether it is translated in English. Can one suggest a way for this- "Online Mathematics ...
6
votes
3answers
775 views

Showing non-cyclic group with $p^2$ elements is Abelian

I have a non-cyclic group $G$ with $p^2$ elements, where $p$ is a prime number. Some (potentially) useful preliminary information I have is that there are exactly $p+1$ subgroups with $p$ elements, ...
1
vote
2answers
153 views

How to find all possible extensions of a finite group by $C_2$

I am not very familiar with group extensions, and I know that in general finding all possible extensions is an extremely hard problem, however I am interested in the following: What are all possible ...
4
votes
1answer
129 views

Is there a general result that groups of order $2^n\cdot 3$ are solvable?

For the past day or so I've been trying to solve an exercise in Lang showing all groups of order less than $60$ are solvable. Excluding the case of order $56$, most cases are taken care of by other ...
1
vote
1answer
143 views

Cocycle expression for the binary tetrahedral group

Since the binary tetrahedral group is an extension of $\mathbb{Z}_2$ by $A_4$, the group product between elements $(h_1,g_1) \cdot (h_2,g_2), h_1,h_2 \in \mathbb{Z}_2, g_1,g_2 \in A_4$ can be ...
3
votes
2answers
141 views

The fundamental group of $K_{3,3}$ — relationship between its generators and embedding into manifolds

So I've been reading this wonderful PDF textbook on algebraic topology: http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf In particular, I'm very interested in the chapter on graphs. This ...
2
votes
1answer
559 views

Largest Normal subgroup

let $G$ be a finite group and $H$ any subgroup. $\psi_{H}$ be the left action of $G$ on $G/H$. It was asked to prove that the action is transitive and the kernel of $\psi_{H}$ is the 'largest normal ...
8
votes
2answers
221 views

Is there a particularly elegant proof that if $|G|=pqr$, then $G$ is solvable?

I know for a fact that if a group $G$ has order $pqr$ with $p,q,r$ distinct primes, then $G$ is solvable. Most proofs I see of this are very ugly, and require a lot of case checking to show most ...
2
votes
0answers
106 views

a question about elements of permutation groups

Let $n \in \mathbb{N}$ and let $S_n$ denote the permutation group on $n$ letters. I'm trying to figure out what kind of elements $\sigma, \delta \in S_n$ satisfy the following relations: ...
1
vote
1answer
95 views

Сondition for the existence of the subgroup

I have an abelian group $G$ of order $m$. And I want to know if there is any subgroup $H$ with order $n$. The condition of the Lagrange's theorem ($m = 0\ (mod\ n)$) seems to be necessary but ...
2
votes
2answers
400 views

If a $p$-Sylow subgroup is in the normalizer of another, they're equal?

I've gotten a little tripped on part of what I'm guessing is a straight forward problem. This is part (a) of Exercise 23 in Lang's Algebra. Let $P,P'$ be $p$-Sylow subgroups of a finite group ...
8
votes
3answers
351 views

Why is closure omitted in some group definitions?

In some texts, there are three group axioms and in some there are four. The difference is that one of the axioms, the closure ($a,b\in G$ then $a*b \in G$) is omitted. Why is this so?
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2answers
422 views

Problem from Armstrong's book, “Groups and Symmetry”

I haven't gotten all that far with this: If $a$, $b$ are members of the permutation group $S_n$, and $ab=ba$, prove that $b$ permutes those integers which are left fixed by $a$. Show that $b$ must ...
3
votes
2answers
123 views

Representation of $S_3$ example in Fulton Harris book

In the Fulton/Harris book "Representation theory: a first course", section 1.3: We are looking at any representation W of $S_3$, by just looking at the abelian group $U_3 \sim Z/3Z \subset S_3$. Let ...
2
votes
1answer
116 views

A question about properties of subgroups $H\subset G$

We say that a subgroup $H\lhd G$ is normal iff it is closed under conjugation by $g \in G$, which implies that for a normal subgroup $gH = Hg$ After reading this definition I wondered, under what ...
4
votes
2answers
988 views

Nonabelian semidirect products of order $pq$?

I just constructed the semidirect product in Lang, and I'm trying to tie some facts together. From Ash's Algebra, I know that if $p\lt q$ are distinct primes, if $q\not\equiv 1\pmod{p}$, then any ...
5
votes
1answer
184 views

Decomposability and irreducibility for representations

I am trying to understand the analogies between linear representations and permutation representations: A representation $\rho:G \to GL(V)$ is irreducible if it has no invariant subspace apart from ...
4
votes
1answer
184 views

Mackey and relatively projective modules

While reading over Alperin's Local Representation Theory and reminding myself how a module is relatively H-projective iff H contains some vertex of the module, I realized I could not prove a basic ...
3
votes
1answer
74 views

Is relatively free the same thing as induced for finite group modules?

I was looking over Alperin's Local Representation Theory and I realized I remembered a definition that may not be there (or true). Is a relatively H-free G-module exactly the same as a G-module ...
1
vote
1answer
224 views

A subgroup containing all the squares. Groups of order 8.

I can not solve the following two problems on Group Theory. A subgroup $H$ of a group $G$ has the property that $x^2 \in H$ for every $x\in G$. Prove that $H$ is normal in $G$ and $G/H$ is abelian. ...
1
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1answer
296 views

Sylow theorems and normalizer

I'll state the result I'm trying to prove, progress I've made, and the two questions I have which will help me solve it. The question is originally motivated by studying defect groups in modular ...
11
votes
1answer
275 views

When can a pair of groups be embedded in each other?

This is a question I made up, but couldn't solve even after some days' thought. Also if any terminology is unclear or nonstandard, please complain. Given groups $G$ and $H$, we say that $G$ can be ...
6
votes
1answer
375 views

Semidirect product uniqueness argument for classifying groups of small order

I'm having trouble understanding the following method for determining the number of semidirect products between two groups in simple cases that arise when trying to classify certain groups of small ...
3
votes
4answers
405 views

Does the operation of $G/A$ on $A$ by conjugation just give a trivial homomorphism?

I'm having doubts about an exercise, since I reach a trivial conclusion. I hope it's not an issue to see if I'm screwing up. Let $G$ be a group, and $A$ a normal abelian subgroup. Show $G/A$ operates ...
3
votes
2answers
156 views

About Finitely Generated Groups

Let G be an infinite group. Suppose that the commutator subgroup [G,G] of G and the abelianization of G are finitely generated. Does this imply that G is finitely generated?
3
votes
1answer
128 views

Minimum Generators of Finite Groups vs. Subgroups

For infinite groups (ex. Free Groups), a subgroup may have more number of generators than the group. For finite groups, is the number of generators of subgroup less than number of generators of ...
4
votes
1answer
667 views

Interpretations of the Correspondence Theorem (4th Isomorphism Theorem)

I know of this version of the correspondence theorem for Groups (From Herstein's Abstract Algebra): ''Let $\phi$ be a group homomorphism from G onto G' with kernel $K$. If $H' \leq G'$ and $H ...
8
votes
1answer
248 views

Quotient group properties

Let $H$ and $E$ be normal subgroups of a group $G$ such that $$G/H \cong E.$$ Under what sort of conditions would we also have $$G/E \cong H?$$ Thanks.
8
votes
3answers
925 views

Three finite groups with the same numbers of elements of each order

There exist pairs of finite groups $G$ and $H$ such that $G$ and $H$ are not isomorphic, yet they have the same number of elements of each order. For example, if $p$ is an odd prime, then the group ...
1
vote
1answer
297 views

Finitely generated subgroups of infinite index

It is well known that if $G$ is a finitely presented subgroup then all its finite index subgroups are finitely presented also. I was wondering if it is possible to extend this, What classes ...
9
votes
2answers
526 views

Free group as a free product

Let $G$ be a group generated by two elements $a$ and $b$. Suppose $G$ is a free group of rank 2. Is it true that $G=\langle a\rangle * \langle b\rangle$? I think the problem is that the definitions I ...
1
vote
0answers
115 views

Showing that all strings representing a group element end the same way

I'm currently working with a group given by the presentation $$\bigl<R,T\,\bigm|\,T^2=(RT)^3=1\bigr>,$$ and I'm trying to prove a result applying to any element with defining string ending in ...
2
votes
1answer
920 views

Determining whether there is a short exact sequence

I'm doing some exercises in Hatcher: Determine whether there is a short exact sequence $ 0 \rightarrow \mathbb{Z}_4 \xrightarrow{f} \mathbb{Z}_8 \oplus \mathbb{Z}_2 \xrightarrow{g} \mathbb{Z}_4 ...
5
votes
3answers
239 views

Is there an elegant way to determine which subgroups of $S_3$ are normal?

I have a homework problem which reads List all subgroups of $S_3$ and determine which subgroups are normal and which are not normal. I understand the definitions of subgroup and normal subgroup, ...
1
vote
2answers
285 views

Finite subgroups of orthogonal transformations in $\mathbb{R}^3$

The group of orthogonal transformations of $\mathbb{R}^3$ is direct product of the group of rotations and the group $Z=\langle x\mapsto -x\rangle\cong \mathbb{Z}/2$. The finite subgroups of group of ...
2
votes
0answers
79 views

Is it true that $\forall G\leq Aut(\mathbb{P}^1)=PGL_2(\mathbb{C})$ the map $\mathbb{P}^1\rightarrow \mathbb{P}^1/G$ is defined over $\mathbb{Q}$?

Is it true that for every finite $G\leq Aut(\mathbb{P}^1_{\mathbb{C}})=PGL_2(\mathbb{C})$ the morphism $\mathbb{P}^1_{\mathbb{C}}\rightarrow \mathbb{P}^1_{\mathbb{C}}/G$ descends as a morphism (not ...
1
vote
1answer
93 views

Clarifications on proof that the fixed points of order $p$, $i_p(G)\equiv -1\pmod{p}$

I'm reading this paper by Marcel Herzog on jstor: http://www.jstor.org/stable/2040939?seq=1 I want to follow up on a few things about the short proof of Theorem 1, found on the bottom of page 1 of ...
1
vote
1answer
205 views

Conjugation / Change of basis in groups

If A is a diagonalizable matrix, then $\exists$ P,D such that $P^{-1}AP=D$. This can be viewed as an inner automorphism of $GL(V)$. More generally, I guess one can write that if $\psi:G \times X \to ...
9
votes
3answers
356 views

Showing $H\unlhd G$ when $[G\colon H]$ is not the smallest prime dividing $|G|$

I recently read about the theorem that for a finite group $G$, if $p$ is the least prime dividing $|G|$, then any subgroup $H$ with $[G\colon H]=p$ is normal in $G$. Going over some exercises, this ...
2
votes
0answers
114 views

What is $\operatorname{Aut}(\operatorname{PSL}_2(\mathbb{F}_q))$? [duplicate]

Possible Duplicate: Automorphisms of projective special linear group I'm sure this is well known, but I don't know where to look up such things. What is ...
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votes
4answers
295 views

Groups of real numbers

In my last question I asked for examples of groups formed by real numbers where the operation is something different from addition or multiplication. With these words I think I could not convey what ...
1
vote
1answer
154 views

Bijection between sets (p-groups and conjugacy classes)

Let $G$ be a finite group. We let $T_{G}$ denote the set of conjugacy classes of subgroups of $G$ and let $T$ denote a set of representatives of $T_{G}$. How to establish a bijection between the ...
4
votes
0answers
186 views

Reference: Geometric group theory

H. Bass has studied existence of lattices on trees. Can someone suggest a (readable) reference for lattices on graphs?