5
votes
1answer
70 views

group-like structure texts.

I was reading Dummit and Foote to be ready for my group theory text, but my teacher seems to be paying special attention to things with less structure than groups, for example monoids, semigroups, and ...
0
votes
0answers
26 views

Question on extension of cocycles

Given a countable discrete group $G$ and suppose $G$ acts on a compact metrizable abelian group $Y$ with normalized Haar measure $\mu$, measure preserving, let $\mathbb{T}$ denotes the unite circle. ...
2
votes
1answer
52 views

Characterization of solvable groups in terms of subgroups of certain orders?

In this question, the OP mentions the following result: a finite group $G$ is solvable if and only if $$\text{for all $n$ dividing $|G|$ such that $\gcd(\frac{|G|}{n},n)=1$, $G$ has a subroup order ...
1
vote
0answers
47 views

Positive definite functions generated by irreducible representations — what do people call them?

Let $G$ be a group and $\pi:G\to B(H)$ be its irreducible unitary representation (one can endow $G$ with topology and claim that $\pi$ is continuous in some sense, this doesn't matter). For a given ...
1
vote
1answer
27 views

Schur Multiplier of general linear group

Ideally I would like to know the Schur multiplier of $Gl(n, F_3)$, but perhaps this is not reasonable to ask. But for a small fixed $n$, this should be known, but i could not find any result when ...
5
votes
3answers
81 views

Pairs of $2\times 2$ matrices generating free groups.

The matrices $\begin{pmatrix}1&2\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\2&1\end{pmatrix}$ are well-known to (freely) generate a free group. Some years ago, I read a paper that, ...
3
votes
1answer
81 views

Textbooks on graph theory

I've read the textbook Groups and Their Graphs by Grossman, and I'm interested in learning more about graphs. I know about O. Ore's book in the same series (Graphs and Their Uses), but I'm interested ...
4
votes
2answers
70 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 ...
7
votes
1answer
84 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. ...
3
votes
4answers
122 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
0
votes
1answer
35 views

The center of Sylow $p$-subgroups of a finite simple group of Lie type

Would some one please to introduce me an easy reference!! which contains the size of $Z(P)$(center of $P$), where $P$ is a Sylow $p$-subgroup of a finite group of Lie type over a finite field of ...
2
votes
2answers
78 views

Sylow $p$-subgroups of finite simple groups of Lie type

I need some information about the Sylow $p$-subgroups, and their normalizers, (specially their sizes), of a finite simple group of Lie type over a finite field (not necessarily algebraic closed) of ...
1
vote
0answers
37 views

Presentations that make the Todd-Coxeter algorithm blow up

Consider the presentation defined, for an integer $n > 1$, by $$G_{n} = \langle x, y \mid xy^n = y^{n+1}x, yx^{n+1} = x^{n}y\rangle .$$ The group defined by this presentation is trivial. Is it ...
0
votes
0answers
22 views

Endomorphisms of Groups - Book Recommendation

Which books dealing with group theory have considerable material on endomorphisms? The books I have seen usually have something on homomorphisms, isomorphisms, and automorphisms, but very little on ...
0
votes
0answers
20 views

Ordinary irreducible representations of semi-direct products

What is the best source for learning about constructing irreducible representations of semi-direct product $G=N \rtimes_\phi H$ from irreducible representations of $H$ and $N$ over field of complex ...
0
votes
0answers
40 views

Request for three foundational papers by Oystein Ore.

Does anybody know how to get or how can I find the following foundational papers on (permutable) groups: O. Ore: Contributions to the theory of groups of finite order (ProjectEuclid Link) O. Ore: ...
3
votes
1answer
57 views

Groups of isometries

I recently read the popular scientific book "Symmetry and the Monster" and it emphasizes groups as the sets of symmetries of geometrical objects. So I was wondering, do all groups appear as symmetry ...
1
vote
3answers
75 views

The commutator subgroup of modular group is a free group of rank 2

In the paper http://projecteuclid.org/download/pdf_1/euclid.ijm/1255632506 it is stated without proof that the commutator subgroup of the modular group is a free group of rank $2$. Can anyone give a ...
0
votes
1answer
80 views

The number of a set of irreducible projective characters vs the number of the ordinary characters of a finite group G.

I need valid references to show that the number of a set of irreducible projective characters with non-trivial factor set is always strictly less than the number of the ordinary characters of a ...
2
votes
1answer
53 views

Classification of parabolic elements of a subgroup of $PSL_2(\mathbb R)$

Let $G\subset PSL_2(\mathbb R)$ be the group generated by the matrices $$a_n=\begin{pmatrix} 1 & 2\cot\frac{\pi}{n}\\0 & 1\end{pmatrix},\; c_n = \begin{pmatrix} ...
3
votes
2answers
104 views

Frattini subgroup of a finite group

I have been looking for information about Frattini subgroup of a finite group. Almost all the books dealing with this topic discuss this subgroup for p-groups. I am actually willing to discuss the ...
0
votes
2answers
31 views

Requested material on Bilinear Pairing

Bilinear map/pairing is widely used in Pairing based Cryptography. I am new to this area. Can anyone suggest me some good reference on Bilinear pairings? I need at least an example of Bilinear map ...
2
votes
0answers
36 views

Inner automorphisms of Lie groups

I have a few questions about $Aut(G)$, when $G$ is a Lie group. It was proven by Hochschild that if $G/G_0$ is finitely generated, then $Aut(G)$ is a Lie group with at most countably many components. ...
4
votes
0answers
179 views

Conjugation Quandles and… “Quandle-Groups”? From quandles to Groups.

A quandle $(Q,*,/ )$ is a idempotent right-distributive and right invertible structure. 1) $a*a=a$ 2) $(a*b)*c=(a*c)*(b*c)$ 3) $(a*b) /b=(a/b)*b=a$ If we have a group $(G, \cdot, ...
2
votes
1answer
25 views

Is the von neumann algebra of locally compact amenable group hyperfinite?

Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite. Does there exist a ...
7
votes
1answer
124 views

Commutator subgroup - or?

If $G$ is a group and $X, Y \subseteq G$ then the commutator subgroup of $G$ is defined as $[G, G] = \langle [x, y] \mid x, y\in G \rangle$, where $[x, y] = x^{-1}y^{-1}xy$ and the group generated by ...
0
votes
0answers
69 views

Describing the sequence A224239.

I've been trying to describe mathematically the $n$th term $a_n$ of the sequence A224239. We get $a_n$ by counting the distinct ways to fill an $n\times n$ grid with squares of smaller integer size, ...
5
votes
1answer
64 views

Reference request: Introduction to Finite Group Cohomology

I don't know anything about group cohomology and I'd like to. What is the best text to learn this subject? I'd prefer as soft an introduction as possible - that is, lots of motivation, lots of ...
2
votes
2answers
83 views

Text on Group Theory and Graphs

A student and I are going to investigate the use of group theoretic techniques in graph theory. What are good texts in this area (introductory and otherwise)? We are particularly interested in ...
0
votes
0answers
17 views

reference that any two elements of $S_n$ are conjugate iff they have the same cycle structure [duplicate]

How can I show/find a reference that any two elements of $S_n$ are conjugate iff they have the same cycle structure.
0
votes
1answer
116 views

Character Table of the Monster

Does anyone have a link to a website having the character table of the Griess Monster in characteristic $0$?
2
votes
0answers
95 views

Calculating sylow subgroups of some concrete groups

Question is to : exhibit all sylow $3$ - subgroups of $S_4$ What i have done so far is : Number of elements in symmetric group is $|S_4|=4.3.2=2^3.3$ number of elements of order $3$ in $S_4$ ...
5
votes
2answers
120 views

What can we say about the kernel of $\phi: F_n \rightarrow S_k$

Let $F_n$ denote the free group on $n$ generators and let $S_k$ denote the symmetric group on the integers $\{1,\dots, k\}$, and the action of homomorphism $\phi$ (as given in the title) on the ...
2
votes
1answer
30 views

Existence of transversals of subgroups implies axiom of choice?

If $G$ is a group and $H\leq G$ is a subgroup, then a transversal of $H$ is a subset $T\subseteq G$ which meets every coset of $H$ in a unique point. The axiom of choice clearly implies that every ...
1
vote
1answer
85 views

What are the conjugacy classes in $\mathrm{Aut}(G)$?

Let $G$ be an arbitrary group, and let $\mathrm{Aut}(G)$ be the group of automorphisms of $G$ (with composition of morphisms as multiplication). I'd like to learn more about the problem of ...
1
vote
5answers
123 views

can somebody recommend a book in a group theory.

can somebody recommend a book in a group theory. that include just questions and their answers. $without$ $theory!$
0
votes
2answers
93 views

What are the good textbooks on Kac-Moody groups?

While there is a number of good books on Kac-Moody algebras ("Infinite dimensional Lie algebras" by Kac is already enough), it seems to me there is lack of textbooks on Kac-Moody groups. nLab says ...
0
votes
0answers
39 views

Definition of K-normal subgroups

"We say that a group K acts semisimply on an abelian group A, if the intersection of all maximal K-normal subgroups of A is trivial" from "Construction of Finite Groups" article by "BESCHE and EICK" ...
2
votes
4answers
161 views

Anyone has a good recommendation of a free pdf book on group theory?

Anyone has a good recommendation of a free pdf book on group theory? I am specially interested in its application for computer science, however, I do not want it to be less mathematically rigorous ...
1
vote
1answer
44 views

Standard reference for Kaluzhnin's theorem

Kaluzhnin's theorem says that if $G$ is a group and $H \leq \operatorname{Aut}G$ acts trivially on each step of the normal series $1 = G_0 \leq G_1 \leq \ldots \leq G_n = G$, then $H$ has nilpotency ...
0
votes
0answers
51 views

Getting free groups of finite/countable rank from certain generating sets

Since I don't know exactly how to explain this I will first describe the idea with the free group on one generator (which I will treat as the integers). Lets say you are given an infinite sequence of ...
2
votes
0answers
37 views

Multiplicative Order Modulo Evaluated Cyclotomic Polynomials

If $\Phi_n(x)$ is the $n$th cyclotomic polynomial, then for which positive integers $n$ and $a>1$ is it true that $\operatorname{ord}_{\Phi_n(a)}(a) = n;$ that is, when is $n$ the smallest positive ...
1
vote
1answer
30 views

Resources on surjunctive groups.

Are there any free available resources on surjunctive groups which are available to say: a graduate level student? A textbook would be fine also. Regards.
1
vote
1answer
95 views

Reference for central product

I am reading central product of groups from text Group Theory I by M Suzuki. But I am neither able to understand nor does getting a feel on what is happening. I will be thankful to you if you can tell ...
0
votes
0answers
144 views

Order of orthogonal groups over finite field

The wikipedia article gives a formula for calculating the order of an orthogonal group over finite filed $O(n,q)$: I don't see how I can get such formula. Can one come up with some references?
0
votes
0answers
19 views

Continuous groups of Transformations [Reference request]

I am considering reading the book : 'Continuous Groups of Transformations' by Luther Pfahler Eisenhart. It seems to have a very interesting table of contents. However this is quite old and I am ...
3
votes
1answer
190 views

Number of groups of order $p^n$, where $p$ is prime

for $n=1$, it is cyclic. so, the number is $1$ for $n=2$, it is Abelian. so, the number is $2$ for $n\geq 3$, I don't know. Can you recommend a book or link which can be helpful for understanding ...
2
votes
3answers
248 views

Can an uncountable group be generated from a single element?

First question : can an uncountable group be cyclic? Ok so my though is if $G$ is generated by i then for $x\in G$ we have $x=i^n$ for integer n, so then it must be countable. Is there a way to ...
1
vote
1answer
114 views

Suggestions for comprehensive maths book library

I've problem that I'm slowly forgetting the math I've learned in early years at university (right now I'm in final year of Mgr. degree as theoretical physicist). I'd like to assemble a finite but ...
5
votes
2answers
75 views

Torsion-freeness of the group $\langle a, b \mid a b^m = ba^n\rangle$

For integers $m$ and $n$ let $K(m,n)$ be the group $\langle a, b \mid a b^m = ba^n\rangle$. Is there a special name for this group? Is there a complete characterization of those pairs $(m,n)$ for ...