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

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Limit set of Kleinian group

Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected ...
2
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2answers
59 views

Why does the additive subgroup of $\mathbb{R}$ generated by $1$ and $\sqrt{2}$ contain arbitrary small elements? [duplicate]

Let $G\subset \mathbb{R}$ be the additive subgroup of $(\mathbb{R},+)$ defined by $G=\mathbb{Z}+\sqrt{2}\mathbb{Z}$. I want to prove that for every $\epsilon>0$ there exists an element ...
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727 views

Masters' thesis in group theory [closed]

I would like some ideas on topics in group theory which would be suitable for a masters' thesis. What sort of problems would be suitable for this level? Because it is at masters' level, no original ...
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2answers
63 views

Probability that two random permutations of an $n$-set commute?

From this MathOverflow question: It is well known that two randomly chosen permutations of $n$ symbols commute with probability $p_n/n!$ where $p_n$ is the number of partitions of $n$. -- Benjamin ...
8
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2answers
143 views

Homogeneous groups

Let's call a group $G$ homogeneous if for every two distinct, non-identity elements $a$ and $b$ there is an automorphism $\phi$ of $G$ such that $\phi(a)=b$. Examining this definition, we can see ...
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86 views

Klein's 4 subgroups

I have just started learning about group theory. And, I learnt about The Klein's 4 group. I tried proving that two distinct Klein's 4 subgroup of a group intersect only at Identity. But I can't. So ...
4
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1answer
89 views

Category theory for graph theory research

I am doing research in algebraic graph theory, focusing on the relation between graphs and groups (especially the representing groups as graphs) for my Ph.D. In particular, one of the ideas is to ...
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84 views

on direct product of two cyclic 2-group

Let $G=C_{2^n}\times C_{2^n}$. Then prove that for any subgroup of $H$ there exist an automorphism that fix only the elements of $H$. explanation: The previous explanation was mistake and for this i ...
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22 views

Homomorphism between $SL(2,\mathbb{C})$ and the Lorentz group and choice of metric

Just a quick question, I'm hoping someone can clarify how this probably small issue can be resolved. It is said that a Lorentz transformation $\Lambda$ is a linear tranformation of $\mathbb{R}^3_1$ ...
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47 views

Normally embedded subgroups reducing in a Hall system

A Hall system of $G$ is a set $\Sigma$ of Hall subgroups of $G$ satisfying the following two properties: -For each $\pi$ divisor if $|G|$, $\Sigma$ contains excatly one Hall $\pi$-subgroup $G_{\pi}$. ...
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1answer
49 views

On decomposition of subgroups a finite abelian groups

Let $G$ be an finite abelian group and $H$ is a subgroup of $G$. Then do can find a decomposition of $G$ as a direct sum of cyclic groups such that the intersections of the summands with $H$ give ...
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65 views

Proving that a group $(G, \ast)$ is abelian if $x^3=x$ for all $x\in G$

If $(G, \ast)$ is a group so that $x^3=x$ for all $x\in G$ then $G$ is abelian
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139 views

Is a finite group always a element-wise product of Sylow subgroups?

Let $G $ be a finite group, and let $ p_1, \ldots, p_n $ be the distinct primes dividing $|G|$. For each $i $, let $ P_i $ be a Sylow $ p_i $-subgroup of $ G $. I seem to recall a theorem saying $ ...
7
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4answers
93 views

Herstein Question: $G^{i}$ normal in $G$?

I just wanted to ask a quick question. I'm going over the second edition of I.N. Herstein's topics in algebra and one of his exercises asks the reader to prove that each $G^{i} $ is a normal subgroup ...
0
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1answer
28 views

Positive-definite function on a group function on a group

I have quite a hard time understanding the definition of positive-definite functions that is based on Hilbert spaces, the one that I read from Wiki; it does not exactly specify that how $H$ relates to ...
4
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1answer
37 views

Is the intersection of the conjugates of a subnormal subgroup of prime power index also a subgroup of prime power index?

I was wondering if it's really the case that, if $G$ is a group with subgroups $H$ and $N$ such that $H\unlhd N\unlhd G$ such that $G/N$ and $N/H$ is a $p$-group, then the intersection of all the ...
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54 views

Group homomorphism $f$ is surjective iff $g$ is

Let $G$ be an additive group, and let $u, v:G\to G$ to be two endomorphisms. Define $f(x) = x- v(u(x))$ and $g(x) = x-u(v(x))$. The question is to show that $f$ is surjective iff $g$ is. I'm only ...
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54 views

The number of orbits of a (normal) subgroup

I want to solve the following problem from Dummit & Foote's Abstract Algebra text: Assume $G$ acts transitively on the finite set $A$ and let $H$ be a normal subgroup of $G$. Let ...
2
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1answer
38 views

Satisfying the hypotheses of the First Isomorphism Theorem.

Let $G$ be the group of invertible upper triangular $2 \times 2$ matrices with real entries and let $H$ be the subset of $G$ consisting of those matrices $A$ with $a_{11} = 1$. It is easy to see that ...
2
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1answer
54 views

Group Axioms Motivation

Group theory is all about symmetries. Can this be seen from the axioms defining a group? Or equivalently can the group axioms be motivated from this point of view? Of course one can look at several ...
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33 views

Showing one to one correspondence

Show that there is a one to one correspondence between the set of left cosets of $H$ in $G$ and the set of right cosets of $H$ in $G$. What is the basic technique/principle for showing one to one ...
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73 views

Methods for counting the number of homomorphisms

Are there any standard methods for counting the number of homomorphisms between groups? For example, how big is $Hom(\mathbb{Z}_2\times\mathbb{Z}_2,D_8)$? My attempt at this was to show that there ...
4
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0answers
53 views

Does there exist an infinite solvable group with no normal abelian subgroups?

This is impossible if $G$ is finite and solvable, because then $G$ has a (nontrivial) minimal normal subgroup $A$, which can be shown (using a trick) to be abelian. I'm trying to mimic the same ...
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36 views

“Bypass” Operations and Groups

So I recently stumbled on this (pdf) collection of group theory related Putnam problems. Problem 1978 A-4 defines a "bypass" operation to be a mapping $\circ:S\times S\mapsto S$ such that $$(w\circ ...
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49 views

Prove that if $H$ is a characteristic subgroup of $K$, and $K$ is a normal subgroup of $G$, then $H$ is a normal subgroup of $G$

$H$ is a characteristic subgroup of $K$ if $\Phi(H)=H~\forall~\Phi\in Aut(K).$ Prove that if $H$ is a characteristic subgroup of $K$, and $K$ is a normal subgroup of $G$, then $H$ is a normal subgroup ...
7
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76 views

Who first proved the fundamental theorem of finitely generated (or finite) abelian groups?

The fundamental theorem of finitely generated abelian groups (or maybe just finite abelian groups) is well-known and can be found in just about any text on the theory of groups or abstract algebra. ...
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54 views

What is the interpretation of $a \equiv b$ mod $H$ in group theory?

I.N. Herstein has defined: Let $G$ be a group, $H$ a subgroup of $G$; for $a,b \in G$ we say $a$ is congruent to $b \mod H$, written as $a \equiv b \mod H$ if $ab^{-1} \in H$. Let $G$ be a ...
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1answer
35 views

An infinite presentation of a group - definition

A finite presentation is a presentation where the set of generators and the set of relators are finite (Source) I'm studying for an exam where one of the "exam topics" is: The example of the infinite ...
3
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2answers
130 views

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known?

Is the order of $\operatorname{Gal}(\bar{\Bbb Q}/\Bbb Q)$ known? And if so is there a description on how the order can be found? My initial thoughts is that because $\bar{\Bbb Q}$ is countable and ...
2
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1answer
20 views

Hopefully easy Lang-Steinberg computation: for Weyl elements

How do you write an element of the Weyl group as $g^{-1} F(g)$? For instance, let $G = \langle x_1(t), x_2(t), x_{-1}(t), x_{-2}(t) : t \in K \rangle$ where $K$ is an algebraically closed field ...
7
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1answer
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Is it possible to partition $\mathbb{N}_+$ into a *finite* family of sets completely not closed under $+$?

Let's say that $A \subseteq \mathbb{N}_+$ is completely not closed under $+$ if $$ \forall_{a,b \in A}[{a+b \notin A}] $$ Is it possible to partition $\mathbb{N}_+$ into a finite family of sets ...
3
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1answer
52 views

$1997$ Putnam - Let $G$ be a group and let $\Phi : G \rightarrow G$ be a function

Let $G$ be a group and let $\Phi : G \rightarrow G$ be a function such that $\Phi(g_1)~ \Phi (g_2)~ \Phi(g_3) = \Phi(h_1)~ \Phi (h_2)~ \Phi(h_3)$ whenever $ g_1~g_2~g_3 = h_1~h_2~h_3=e$ . Show ...
3
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1answer
65 views

Permutations: If I know $\alpha$ and the cycle structure of $\alpha\beta$, can I find $\gamma$ for which $\gamma\beta$ also has this cycle structure?

Suppose we have two permutations $\alpha$ and $\beta$ (of a set $S$ of size $|S|=n$), and I know $\alpha$ and the cycle structure of $\alpha\beta$. But I don't know $\beta$. Can I find a ...
3
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3answers
133 views

On the Importance of the Second and Fourth Isomorphism Theorems

I suppose I'd like to focus on the theorems for groups and rings, first of all. In particular, I'd rather not see anything about modules, simply because I don't feel I know enough about them. Anyway, ...
4
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2answers
80 views

Error in Hungerford's algebra proof? Left id & inv = group

Prop 1.3 in Hungerford's Algebra said that if $G$ is a semigroup and there exist a left identity and each element have a left inverse, then $G$ is a group. The proof (and in fact, even the proposition ...
2
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2answers
48 views

Homomorphism $f: \mathbb{C}^{*}\rightarrow \mathbb{R}^{+}$. Prove that kernel of f is infinite group.

First of all we need to prove that $\ker(f)$ is group by proving: That $\ker(f)$ contains $e\in\mathbb{C}^*$, That $\ker(f)$ is closed under multiplication for every $a,b \in \ker(f)$ That ...
0
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1answer
33 views

A group which multiplies input by 2

I want a group in $Z$ which multiplies the input by 2. I'm considering a group in $Z$ with the operator $2a$ where $a \in Z$ but I'm a bit rusty in abstract algebra. Can I use such an operator?
0
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1answer
26 views

Direct Product of Torsion Subgroups

So I came up with this theorem while studying, and concocted a small proof, and I was wondering if someone could verify it, as I am very new to torsion groups/elements. I am open to all criticism. ...
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1answer
38 views

A group with bounded element orders and its minimal and maximal subgroups.

Let $n>1$ be an integer. Is there an abelian group $G$ with all elements of order less than $n$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains ...
1
vote
1answer
29 views

Promises in the hidden subgroup formulation of graph isomorphism problem

In the 3rd slide of the talk, Graph isomorphism, the hidden subgroup problem and identifying quantum states, by Pranab Sen, the promise of the problem is defined as follows. Given: $G$: group, ...
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0answers
32 views

identifying a subgroup of $S_8$ generated by 4-cycles

Let $G \subseteq S_8$ be the subgroup generated by some 4-cycles. If we number the elements $1,2,\dots, 8$, the 4-cycles are $(1234),(5678),(1485),(2376),(1265),(4378)$ I am not sure if I have ...
2
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1answer
48 views

0n group that have an non-trivial element fix with each automorphism

Let $G$ be a group and $Aut(G)$ is an automorphisms groups of $G$. We know that if $Aut(G)$ is nilpotent and $G$ is not cyclic of odd order, then $G$ has an non-trivial element such that fix by all ...
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0answers
15 views

maximal and minimal subgroups of torsion abelian groups

Is there a torsion abelian group $G$ for which exactly one of these conditions is correct: 1) every non-trivial subgroup of $G$ contains a minimal (non-trivial) subgroup of $G$. 2) every proper ...
4
votes
1answer
60 views

Does $\displaystyle \frac{G}{H}$ $\simeq$ $\displaystyle \frac{G}{K}$ $\Rightarrow$ $H$ $\simeq$ $K$?

Does $\displaystyle \frac{G}{H}$ $\simeq$ $\displaystyle \frac{G}{K}$ $\Rightarrow$ $H$ $\simeq$ $K$? I think it's true but I am having trouble demonstrating it. If $H$ and $K$ are subgroup of a ...
2
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1answer
33 views

tree structure on classes of elements in GL_2 over a field with discrete valuation

this is my first question here, so I hope I am doing it right. :) I'm currently reading a paper about the tree of GL_2 over a discretely valued field (similarly to Serre). Here's the setting: $k$ an ...
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1answer
57 views

Prove : Aut(G) is abelian so G is cyclic [closed]

Let $G$ be a finite group. Suppose $Aut (G)$ is abelian: prove that $G$ is cyclic .
2
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1answer
55 views

Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$?

Question as stated in the title: Has a finite group, generated by $a,b$ always a relation of the form $1=a b a^{\alpha} b^{\beta}$? If not, can you give me a counterexample? Thanks
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1answer
41 views

Show that $|Aut(G)|<n^{\log_2(n)}$ where $G$ is finite

Let $G$ be a finite group of order $n>1$. Show that $|Aut($G$)|<n^{\log_2(n)}$. The 'only' thing I know is the group of inner automorphisms is isomorphic to $G/Z(G)$ and by definition, ...
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2answers
37 views

Cyclic groups and Lagrange's Theorem

This may seem somewhat of an easy proof, but as I'm currently working through a sample paper, I'll definition be getting all my answers checked here (if I cannot find solutions online). Question: ...
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3answers
37 views

Does represented ring appear to be a field? [closed]

$\mathbb{R}[x]/(x^{2}+1,x^3-2x^2+x-2)$ Hello! My name is Ramzan! I`m solving this issue!