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

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5
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107 views

How much does $\operatorname{Aut}(H_1(S))$ determine a homeomorphism $S \to S$?

Let $S$ be an orientable compact surface. A homeomorphism $f: S \to S$ induces an isomorphism $f_{*}: H_1(S) \to H_1(S)$. How much can we say the converse? Namely, if we are given an element of ...
2
votes
3answers
209 views

Question about Cyclic Group

$(G,\star)$ is a $n$-order cyclic group, the generator is $x$, Prove: 1) For any factor $d$ of $n$ exists only one $d$-order subgroup. 2) $b=x^k$ is $G$'s generator if and only if ...
3
votes
0answers
217 views

On Sylow subgroup of simple group PSL(2,p)

Let $G$ be finite group such that $Z(G)=1$ and the number of Sylow $p$-subgroups of $G$ equal to the number of Sylow $p-$subgroups of $PSL(2,r)$ for every prime $p$ ( $r$ is prime and $r^2$ not ...
2
votes
2answers
333 views

Finite extensions of groups

Let $G$ be a group which is a finite extension of the group $H$. As far as I know, by definition, this means that there is a finite normal subgroup $N$ of $G$ such that $G/N=H$. But, is it ...
6
votes
1answer
232 views

Galois and solvable primitive permutation groups

I posted on this subject recently, but there was a misunderstanding on my side. Since my French is not very good, I misread the Galois's original paper. So let me explain my question again. Let ...
3
votes
1answer
195 views

If $G$ has a normal subgroup of order 2 and infinite cyclic quotient, $G$ is abelian?

Assume that $G$ has a normal subgroup $H$ of order $2$ (isomorphic to $Z_{2}$) and $G/H$ is infinite cyclic (which indicates that $G$ is also infinite order). The target here is to prove that $G$ is ...
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vote
1answer
43 views

Lemma about f.g. abelian groups endowed with a certain bilinear form

Let $G$ be a finitely generated abelian group (written additively), let $\langle \cdot,\cdot\rangle\colon G\times G\to \mathbb Z$ be a bilinear form and let $\sigma\colon G\rightarrow G$ be an ...
12
votes
1answer
433 views

Prove that $G$ is abelian

Let $G$ be a group with the property that for any set of three distinct elements in $G$, say $x$, $y$, $z$, at least two of them will commute. Prove that $G$ is abelian. I have no idea how to ...
2
votes
1answer
220 views

Finite-by-(abelian-by-finite)

I want to prove that a (finite-by-abelian)-by-finite is also finite-by-(abelian-by-finite). If we have a (finite-by-abelian)-by-finite group $G$ that means that it has a normal subgroup of finite ...
3
votes
1answer
114 views

Quotients of virtually abelian groups

If $G$ is virtually abelian, is it true that any quotient of $G$ is? If it is true can you give me a proof?
2
votes
2answers
396 views

Categories and the Isomorphism Theorems

I've been trying to work through Mac Lane's "Categories for the Working Mathematician" on my own, but I seem to be struggling with the concept of universality (arrows and elements). In particular, I ...
1
vote
1answer
165 views

Why all the elements of sylow subgroups are not adding up to the no. of elements of Group.

Show that a group of order 70 can not be simple. I've tried to solve using Sylow theorem. I got 1, 5, 7, 35 Sylow 2-subgroups, 1 sylow 5-subgroup and 1 sylow 7-subgroup. Now the only choice is 35 ...
3
votes
2answers
222 views

The index of a subgroup is divisible by the index of its image under a homomorphism

Let $f$ be a homomorphism defined on a finite group $G$, and let $H$ is the subgroup of $G$. Then show that $$ \left [f(G) : f(H)\right] \text{ divides } \left [G : H\right].$$ I know $$\left [G : ...
5
votes
3answers
461 views

Platonic Solids

It´s a theorem that there exist only five platonic solids ( up to similarity). I was searching some proofs of this, but I could not. I want to see some proof of this, specially one that uses ...
5
votes
3answers
3k views

Does the intersection of two finite index subgroups have finite index?

Let $(G,*)$ be a group and $H,K$ be two subgroups of $G$ of finite index (the number of left cosets of $H$ and $K$ in $G$). Is the set $H\cap K$ also a subgroup of finite index? I feel like need that ...
2
votes
0answers
139 views

Galois's statement on a solvable primitive permutation group

The following statement is equivalent to the one Galois wrote in a paper submitted in 1830. Is this correct? Let G be a finite solvable group acting faithfully and primitively on a set S. If a, b are ...
6
votes
1answer
180 views

Direct products of infinite groups

By an argument which is not entirely trivial we know that if $G,H$ are finite groups and $G \times G \cong H \times H$, then $G \cong H$. I was wondering if this result holds if $G, H$ are infinite, ...
0
votes
1answer
96 views

$S_7$ does not contain a subgroup of order 15

$S_7$ does not contain a subgroup of order 15 This is an example of converse of Lagrange's Theorem not working. I want to know how we can prove this. Do you know which idea we should use?
2
votes
4answers
245 views

Isomorphism of symmetric groups.

True or false? Give reason. $S_m\times S_n\simeq S_{m+n}$. I know this is not true but I don't know how to prove it.
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vote
1answer
500 views

Group homomorphism always exist between two groups.

Is there always exists a homomorphism between two groups $G_1$ and $G_2$? Why?
4
votes
2answers
244 views

Understanding the semidirect product

I'm trying to motivate the definition of a semidirect product, and it seems like it comes from geometric examples. Some classic ones are $D_{2n} \cong Z_n \rtimes Z_2$ $E(n) \cong R^2 \rtimes O(2)$ ...
6
votes
1answer
585 views

Semidirect Products with GAP

I'm wondering how to specify to GAP which homomorphism to use when constructing a semidirect product. I'm trying to have it construct $\left(\mathbb{Z}_p\times\mathbb{Z}_p\right)\rtimes_\varphi S_3$. ...
3
votes
1answer
140 views

Why is the group of covering transformations relative to the quotient map isomorphic to a subgroup of the Fundamental Group?

I'm trying to prove the classification theorem for covering spaces. I've got to the stage where I need to show the following: If $H$ a subgroup of $\Pi_1(X,x_0)$ then $\exists Y$ covering space of ...
3
votes
1answer
219 views

abelian transitive subgroups

Can anybody tell me what is known about the classification of abelian transitive groups of the symmetric groups? For instance: Let $G$ be a an abelian transitive subgroup of the symmetric group ...
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vote
0answers
104 views

Hall subgroup of a finite group G

If $G$ is a finite group and $x$ is a $p'$ element of G does this imply that there a Hall $p'$ subgroup of $G$ containing $x$?
2
votes
1answer
144 views

Find the structure of $\mathbb{Z}[\sqrt[3]{2}]/(4+\sqrt[3]{4})$

Let $A=\mathbb{Z}[\sqrt[3]{2}]$ and $I=(4+\sqrt[3]{2^2})$. Elements in $A$ have the form $a\cdot 1+b\cdot 2^{\frac{1}{3}}+c\cdot 2^{\frac{2}{3}} \Rightarrow$ elements in $I$ have the form $$ (a\cdot ...
0
votes
1answer
212 views

If a group has elements of order 25 and 49, then it has an element of order 35

Let $a,b$ be elements in a group $G$ such that $|a|=25$ and $|b|=49$. Prove that $G$ contains an element of order 35. If $G$ is finite then we can say that $5,7 \mid G$ and hence by Cauchy's ...
3
votes
1answer
132 views

G acts primitively, faithfully on A. |A| is even, show |G| is divisible by 4

I want to ask for a hint to this problem: G acts primitively, faithfully on A. |A| is even, show that |G| is divisible by 4
10
votes
5answers
411 views

Definition of group action

I'm currently taking a class in abstract algebra, and the textbook we are using is Ted Shifrin's Abstract Algebra: A Geometric Approach. In the chapter on group actions and symmetry, he defines a ...
7
votes
3answers
414 views

Minimal axioms for a group

My group theory is very rusty. If I want to just start with left inverses and left identities, must I link the axioms, or can I leave them independent? e.g., is it enough to say "there exists at ...
2
votes
1answer
115 views

Relationship between rose and categorical definition of a group?

Given a free group $F_X$, and it's Cayley graph, $Cay(F_X,X)$, we can take the quotient graph of $Cay(F_X,X)$ by the action of $F_X$ on it and obtain $R_{|X|}$, a "bouquet of circles" or "rose" ...
5
votes
1answer
417 views

Burnside's Theorem

I have seen two statements of Burnside's Theorem and they are as follows. Statement 1: Let $p, q$ be distinct prime numbers and $a,b \in \mathbb{Z}_{\geq 0}$. There does not exist a non-abelian ...
4
votes
1answer
154 views

Free groups and Kazhdan's property (T)

Showing non-amenability of a (non-abelian) free group is somewhat easy and one can do this immediately after the definition of amenability. Is there an easy proof of the fact that free groups do not ...
3
votes
1answer
87 views

two subgroups of $S_{n}$ and $S_{m}$

If $H\subseteq S_{n}$ and $K\subseteq S_{m}$ how can I then show that I can think of $H\times K$ as it was a subgroup of $S_{m+n}$?
7
votes
0answers
273 views

What groups are semidirect products of simple groups?

The question I want to ask is actually slightly broader than that in the title: what is the smallest class of finite groups which contains all finite simple groups groups, and is closed under ...
6
votes
2answers
747 views

Infinite nilpotent group, any normal subgroup intersects the center nontrivially

Having problems with the following statement: If $N$ is a nontrivial normal subgroup of a nilpotent group $G$ then $N \cap Z(G) \neq \langle e \rangle$ Here $Z(G)$ denotes the center of $G$. ...
3
votes
0answers
158 views

About a Sylow subgroup of a product

Let $G$ be a finite group, $H$ and $K$ subgroups of $G$ such that $G=HK$. Show that there exists a $p$-Sylow subgroup $P$ of G such that $P=(P\cap H)(P\cap K)$. I looked at the proof here, but I ...
3
votes
3answers
264 views

$(A\times B)\cap {(C\times D)}=(A\cap C)\times(B\cap D)$ Analogy in Group Theory

In set theory we have $(A\times B)\cap {(C\times D)}=(A\cap C)\times(B\cap D)$, is this true in group theory if we replace the $\times$ by products of subgroups i.e. $$(AB)\cap (CD)=(A\cap C)(B\cap ...
4
votes
3answers
241 views

What is the group generated by the conjugacy class containing $(12\ldots n)$ in $S_n$?

This question is motivated by this answer to a question about groups generated by conjugacy classes. Let $n \geq 1$ and $S_n$ be the symmetric group on $\{1,2,\ldots,n\}$. Define the n-cycle ...
3
votes
0answers
135 views

Terminology in Dickson's book

This appears in Dickson's "Linear Groups with an exposition of the Galois field theory", page 50, chapter 4. My question is: how would the above "translate" in modern terms? In particular, do we ...
0
votes
1answer
68 views

Regular normal forms of two groups

If ${\bf G}$ and ${\bf H}$ have regular normal forms, then so does ${\bf G}$ x ${\bf H}$ and ${\bf G}$ * ${\bf H}$. I am new to the concepts of regular normal forms. If anyone could offer any ...
0
votes
3answers
955 views

Dihedral group - elements of order $2$

If $D_{n}$ is the Dihedral group of degree $n$ (order $2n$) where n is an odd number, can I then conclude that the numbers of elements of order 2 in $D_{n}$ is equal $n?$ I suppose there are the ...
2
votes
1answer
86 views

A finite simple group with no more than 6 p-Sylow subgroups for any $p$ is cyclic [duplicate]

Possible Duplicate: Proving a theorem about a finite simple group My homework: Let $G$ be a finite simple group such that for any prime $p$, the group $G$ has at most 6 $p$-Sylow subgroups. ...
5
votes
2answers
126 views

Representation in $A_8$ of $D_4$ allows automorphism that switches 2 normal subgroups

Name the corners of a square as 1,2,3,4 in clockwise order. As you know: The group of all rigid motions of the square back to itself, called $D_4$, has eight elements, written in cycle form as: ...
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2answers
485 views

Groups and generating sets

This question feels completely trivial and I am somewhat embarrassed to be asking it, but I am having a brain dead moment and failing to prove what I'm sure is a completely trivial statement about ...
2
votes
2answers
152 views

Number of transitive acting groups on four letters?

Constructing a group (a permutation group) which acts on a set of 4 letters transitively is easy. for example $G_{1}=\{id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)\}$ < $S_{4}$. Can it be verified how ...
3
votes
2answers
711 views

Intersections of all subgroups is a nontrivial subgroup, so every element has finite order.

I need help to proof this result: "Let G be a group such that the intersection of all its subgroups other than $\{1\}$ is a subgroup different than $\{1\}$. Then all its elements have a finite ...
4
votes
1answer
274 views

Order of a normal subgroup

I need help showing this result: "Let $G$ be a group such that $|G|=nm$ where $m$ and $n$ are relatively prime. Suppose that there exists a normal subgroup H of G such that $|H|=n$. Show that $H$ is ...
5
votes
1answer
103 views

Is there a “natural” transitive action of $SL_2(\mathbb{F}_5)$ on a set with 5 elements?

I'm really looking for a "cute" way of showing that $SL_2(\mathbb{F}_5)$ is a double cover of $A_5$. The sort of action I am looking for is something like the action of $GL_2(\mathbb{F}_3)$ on ...
2
votes
1answer
126 views

Vertex of a trivial source module

This should be easy for the resident representation theory specialists: Let $F$ be an algebraically closed field of characteristic $p>0$, $G$ a finite group, and $M$ an indecomposable $FG$-module ...