A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Significance of deletion and exchange conditions in reflection groups

I am having trouble warping my head around the exchange and deletion conditions in finite reflection groups (i.e.Coxeter groups). It is mentioned as the "characterising property of coxeter groups ...
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78 views

Computing $\mathbb{C}[x,y]^G$ or $\mathbb{C}[x,y,z]^G$ where $G$ is a finite subgroup of $GL_n(\mathbb{C})$

My question is related to this link: Ring of Invariant $\mathbf{Question \;1}$. Let $$ A = \left( \begin{array}{cc} 0 & -1 \\ 1& 0 \\ \end{array} \right). $$ Then $C= \langle A\rangle$ ...
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51 views

Jacobian of group products

I need to find the Jacobian of a product of SE(3) group transformations: $G = X*Y*X^{-1}$ Help appreciated!
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1answer
82 views

Central divisible subgroup

Have you any nice example of central divisible subgroup of a finitely presented group ? (of course the subgroup has not to be trivial)
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Does every embedding $\varphi:\mathrm{UT}(n,\mathbb{R})\to\mathrm{UT}(m,\mathbb{R})$ extend to $\bar{\varphi}:T^*(n,\mathbb{R})\to T^*(m,\mathbb{R})$?

EDIT: I've now posted this question on MathOverflow. Here $T^*$ means upper triangular with positive diagonal entries and $\mathrm{UT}$ means upper triangular matrices with all diagonal entries ...
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78 views

Is $P_\omega$ a $p$-Sylow subgroup of $G_\omega$

We have the following theorem Let $G$ be a group, acting on a set $\Omega$ and let $p^m\Bigm||\omega^G|$ wherein $p$ is prime and $\omega \in \Omega$. If $P$ is a $p$-Sylow subgroup of $G$ then, ...
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89 views

Thompson's group F and monoidal categories

Fiore and Leinster have proved that if $\mathcal{A}$ is a free monoidal category generated by one object $A$ such that there exists an isomorphism $\alpha: A \otimes A \to A$, then for every object $X ...
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64 views

Plancherel and characters

Let $G$ be a locally compact abelian group (non compact) and $\mu$ a Haar measure on $\hat{G}$ given by the Plancherel Theorem. Suppose $f\in L^2(G)$. I would like to write a formula as $$ ...
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1answer
516 views

What can we say about functions satisfying $f(a + b) = f(a)f(b) $ for all $a,b\in \mathbb{R}$? [duplicate]

Possible Duplicate: Is there a name for such kind of function? I am investigating functions satisfying the exponentiation identity $f(a + b) = f(a)f(b)$ for all $a,b\in \Bbb R$. This is ...
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1answer
197 views

Troubling calculation for a wreath product of groups.

Let $G$ be a group, $H$ a transformation group acting on a set $S$, and suppose $G$ acts on another set $T$. Let $G\wr H$ denote the wreath product of $G$ and $H$. So the composition for ...
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131 views

Structure of elementary abelian group by a cyclic group of coprime order

My mind is a little foggy right now, so I would like to ask for some help on this (or an appropriate reference). Suppose $A$ is an elementary abelian $p$-group and $B$ is a cyclic group of order ...
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76 views

JSJ-decompositions of groups and 3-manifolds: a reference request

I am, for whatever reason, interested in learning about the JSJ-decomposition of groups. Having asked around a bit, it was suggested I first learn about what is happening in the manifolds and then ...
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88 views

How do I construct the $\operatorname{SU}(2)$ representation of the Lorentz Group using that $\text{SU}(2)\times\text{SU}(2)\cong \text{SO}(3,1)$?

This question is so mathematical that I think I'll have more luck asking it in the mathematics section, than I would in the physics section. This is problem II.3.1 in Anthony Zee's book Quantum Field ...
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1answer
104 views

Subgroups of free products

Let $G$ be a free product of 2 groups with $G \neq Z_2*Z_2$ . i would like to know if the following assertion is correct : Every almost nilpotent subgroup of G is contained in a unique maximal almost ...
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372 views

All irreducible representations of Pauli group

I'm supposed to find out all irreducible representations of Pauli group, that is, the group generated by Pauli matrices $\sigma_k(k=1,2,3)$. It has 16 elements: $\pm 1, \pm i, \pm \sigma_k, \pm i ...
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109 views

a computation in Galois theory

I am confused with some computation in Galois theory (this is not homework, just my weird curiosity). Let $k$ be a field of positive characteristic $p\neq 2$ that contains all roots of unity (e.g. ...
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481 views

A subgroup of symmetric group of degree n with index smaller than n.

When I read an article about Cauchy's theorem (1815) on a permutation group, I tried to prove it and I came up with the following proposition which is similar to the Cauchy's but is more general. Is ...
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244 views

Wreath Product of Two Finitely Generated Groups

Let $G$ and $H$ be two finitely generated groups, and let $W = G \wr H$ be the wreath product of $G$ and $H$. Show that $W$ is finitely generated. In class today, we were showed this and told that it ...
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102 views

Finding elements of specific order in $GL_n(\mathbb{Z}/2\mathbb{Z})$

Consider $G=GL_n(\mathbb{Z}/2\mathbb{Z})$. What is the smallest $n$ for which $G$ has an element of order 8? Give an example of an element of order 8. I've thought about just considering what happens ...
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148 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 ...
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144 views

Classification Theorem Of Abelian Groups-Question regarding the proof

I'm currently reading Munkres-Algebraic topology text, and in his review chapter of abelian groups, he gives the classification theorem for finitely generated abelian groups. He ommited some important ...
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55 views

Dehn Twist in the sense of graphs

Does anyone knows a good book or script about Dehn Twists in the sense of graphs. More precisely: I need to know how a Dehn Twist yields an automorphism of a group or subgroups. I want to know ...
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235 views

On automorphisms of a Frobenius group

Consider the following facts: A Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a ...
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150 views

Are there 16 or 24 automorphisms of $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$?

In this question I said that the automorphism group of $\mathbb{Z}/4\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}$ has 16 elements because If $\varphi$ is one of this automorphism then ...
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116 views

What is the easiest example of a finitely presented group which is not residually finite?

What is an easy example of a finitely presented group which is not residually finite? To be clear, part of the question is: how do we see that it isn't?
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57 views

For a $G$-module $A$ is there a maximal subgroup $H$ of $G$ such that the image $H^2(G,A)\rightarrow H^2(H,A)=0$?

Let $G$ be a group, and let $A$ be a $G$-module. Then for every subgroup $H$ of $G$, $A$ is also an $H$-module. Furthermore, there's a map $H^2(G,A)\rightarrow H^2(H,A)$. I would like to know ...
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143 views

Hirsch length of polycyclic groups

I've got the following exercise to solve: Let $G$ be a polycyclic group, $N \lhd G$ and let $h(\cdot)$ be the Hirsch length. Then $h(G) = h(N) + h(G/N)$ I know that subgroups and quotients of ...
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175 views

Determinants and homomorphisms of general linear groups

Consider the functions $\rho_1:M_1(\mathbb C)\to M_2(\mathbb R)$ where $$\rho_1(a+bi)=\begin{pmatrix} a&b\\ -b&a \end{pmatrix}$$ and $\rho_2:M_2(\mathbb C)\to M_4(\mathbb R)$ where ...
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65 views

having a subgroup of $\mathbb{Z}^{3}$ and wants to show linear independency with parameters

I am stuck with these hard-star exercises for some time now (they are from a book called "Introduction a L'Algebre et L'Analyse Modernes" de M.Zamansky"), if somebody sees the right way, I will be ...
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87 views

Induced representations of topological groups

Sorry if this is a naive question-- I'm trying to learn this stuff. If $G$ is a group with subgroup $H$, then we have the restriction functor $\operatorname{Res}$ from $G-\operatorname{mod}$ to ...
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137 views

A subgroup of $\operatorname{sp}(4,\mathbb{Z})$

Let $M_1$ and $M_2$ be two matrices as follows: $ M_1= \left( \begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 5 & 5 & 1 & 0 \\ 0 & -5 & ...
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164 views

If $G$ is a finite group and $(ab)^3= a^3b^3$, [duplicate]

Possible Duplicate: Group with an endomorphism that is “almost” abelian is abelian If $G$ is a finite group and $(ab)^3= a^3b^3$, and $3 \nmid o(G)$, then how do I prove that ...
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113 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: ...
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80 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 ...
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118 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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300 views

Finite subgroups of PGL(2,K)

Can one give an elementary proof of following interesting theorem? (These are from the paper "Finite subgroups of $PGL(2,K)$-Beauville"). (In each statement , characteristic of $K$ is prime to order ...
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227 views

Isomorphism of groups with certain property, p-groups

I'm doing exercises from Hungerford's book "Abstract Algebra: An Introduction". The exercise is in section 8.2, numbered 22. I would like someone to check my proof, as I have reasonable doubts that ...
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181 views

An abelian group with $n$ generators and $r(r<n)$ more relations is infinite

Let $A$ be an abelian group with generators $x_1,x_2, \cdots, x_n$ and defining relations conssisting of $[x_i,x_j]$, $i<j=1,2, \cdots, n$, and $r$ further relations. If $r<n$, prove that $A$ ...
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366 views

Irreducible finite dimensional representations of $SL(2,\mathbb C)$

Is there a book that finds all the irreducibile finite dimensional representations of $SL(2,\mathbb C)$ without considering the Lie algebra $sl(2,\mathbb C)$? For example, how can I show "directly" ...
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131 views

Property (T) for groups vs top

I have encountered two properties in different areas of math. One is the property (T) of groups and the other is the property (T) of topologies. What is the connection between these two ? Thank you. ...
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181 views

A question about reduced torsion abelian groups

If a reduced torsion abelian group has no cyclic direct summands of order greater than 2, is it an elementary abelian 2-group? Background: I'm trying to classify the groups whose group rings have a ...
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212 views

On amalgamated free product

If $G$ is a group, with epimorphism $\phi \colon G\rightarrow H$, and if $H=H_1*_{H_3}H_2$ for $H_i\leq H$, then is it true that $G=G_1*_{G_3}G_2$, where $\phi(G_i)=H_i$? (If yes, how? If not, ...
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88 views

Every subnormal of a semisimple is normal

There is a famous lemma saying: "Every subnormal subgroup of a semisimple group is normal and is a semisimple direct factor of the group". Any hints about it? :)
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62 views

Subgroups Lattice of Automorphism Group of Linear Groups

Could someone tell me the subgroups lattice of $PSL(n,p^k)$ or could someone tell me sources I should read to know all about $Aut(PSL(n,p^k))$ and its subgroups lattice. Thanks in advance.
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131 views

Representation theory question + SU(n)

Would you please help me in how to solve these questions : A- Let $H$ be a subgroup of a finite group $G$.Let $\alpha$ and $\beta$ be class function of $G$ and $H$ respectively. Prove that ...
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186 views

A question on a generalization of perfect numbers

First of all, I would like to call a group immaculate provided that the orders of $G$ and $\Sigma$ (the order of $N$) where $N$ varies over all normal subgroups of $G$, are equal. From here it has ...
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218 views

Involution centralizer does not determine the group

Is there a finite group which is contained in infinitely many core-free finite groups as the centralizer of a central involution? The Brauer–Fowler results show that if a finite group has no ...
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1answer
60 views

Some questions about Banach Tarski proof

Banach-Tarski proof as been the topic of a video by the well-known Youtube channel VSauce but there were some parts that I didn't understand. So I went reading for the proof on Wikipedia, and I didn't ...
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1answer
42 views

Help proving inequality

3. $\delta$ is the standard Euclidean valuation on $\mathbb{Z}[i]$. For each of the following pairs $a,b \in \mathbb{Z}[i]$, find $q,r \in \mathbb{Z}[i]$ such that $a = qb + r$, where either $r = ...
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2answers
73 views

What can we say about the order of a group?

Let $G$ be a group and $a ∈ G$. If $a^{12}= e$, what can we say about the order of $a$? What can we say about the order of $G$? We know that $|a|$ divides $12$, but what can we say about the order of ...