0
votes
2answers
24 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
33 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
135 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
20 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
108 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
65 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
43 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
58 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
72 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
76 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
102 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
28 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
79 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
102 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
74 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
37 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
131 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
40 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
45 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
33 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
28 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
65 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
101 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
17 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 ...
2
votes
1answer
152 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
193 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
96 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
72 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 ...
12
votes
2answers
286 views

Are there Groups of Strictly Primes

Motivation Since Euclid's proof of the infinitude of the primes, the structure and properties of primes has always fascinated mathematicians. This lead to great work in their properties and ...
5
votes
1answer
159 views

When are two semidirect products isomorphic?

Let $N$, $H$ be groups, and $\varphi : H \to \operatorname{Aut}(N)$ a group homomorphism. Then we can form $N \rtimes_{\varphi} H$, the semidirect product of $N$ and $H$ with respect to $\varphi$. ...
1
vote
1answer
84 views

Can you give me a good alternative to Rotman's Group Theory book?

I've been trying to learn out of Rotman's book "An Introduction To The Theory of Groups" for the last few months, and it's rough going. I've been studying Chapters 7, 10, and 11 in particular, and ...
2
votes
3answers
920 views

Video lectures on Group Theory

The web is full of video lectures these days but, try as I might, I can find very little for Introduction to Group Theory. The closest I found was ...
3
votes
0answers
140 views

How “bad” can presentation of the trivial group get?

These questions are sort of preliminary questions and reference requests for a project I am doing. Lets say, for concreteness, that $R$ is a set of words in the free group of rank two and that ...
0
votes
0answers
50 views

reference for proof a theorem in Group Theory

Let $G$ be a group and $K$ be a subgroup of $G$ of index $2$ ($|G:K|=2$). Let $H$ be a normal subgroup of $G$. Then H is subgroup of K or $|K:K \cap H|=2$. I want a reference for see the proof of this ...
1
vote
0answers
47 views

Are Heisenberg groups modulo $p$ 2-closed?

Have anyone ever published that the Heisenberg groups modulo $p$ ($p$ prime) are 2-closed groups? If yes, where can I find it? (A permutation group $G$ is 2-closed, if the full automorphism group of ...
8
votes
1answer
93 views

$\operatorname{Aut}(G)$ contains an involution $\sigma$ with no nontrivial fixed point

I am just reading some algebra books on my own, and it seems the following exercise appears in so many of them: Let $G$ be a finite group with $\sigma\in\operatorname{Aut}(G)$ satisfying ...
2
votes
0answers
84 views

What are the one-parameter subgroups of GL?

Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms $\lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk$ of algebraic groups) completely classified? The obvious ...
4
votes
1answer
56 views

reference request: proof that group characters are a basis for $L^2$

I know the following must be very standard, but I haven't found it in any of the functional analysis books to which I have access. Do you know where I can find a self-contained proof?: If $G$ is a ...
-1
votes
1answer
101 views

About a past question [closed]

In What is the center of a semidirect product, $\operatorname{Z}(G_1 \rtimes_\varphi G_2)$?, Alexander Gruber answered a question of user39794. Is there cited reference for this? I want to cite this ...
6
votes
1answer
132 views

How to Find Maximal Abelian Subgroups?

I am studying some groups (they are infinite, finitely represented, nilpotent) and am trying to find their maximal abelian subgroups. Is there any standard approach to do so? Can anyone recommend any ...
3
votes
0answers
87 views

Classification of Groups

Arthur Cayley classified all groups of order $4$ and $6$ in 1854, and groups of order $8$ in 1858. What about groups of order $2,3,5,7$?. These are prime numbers, and the most basic theorem in group ...
3
votes
1answer
88 views

The structure of $(\mathbb Z/525\mathbb Z)^\times$

I am working on the following problem. Find the number of the elements of order 4 in $(\mathbb Z/525\mathbb Z)^\times$. I tried to solve it in the following way: since $525=3\cdot5^2\cdot7$, we ...
3
votes
0answers
84 views

Are simple algebraic groups absolutely simple?

Let $k$ be a field. By a simple algebraic group over $k$ I mean an affine group scheme $G$ of finite type over $k$ such that $G$ is connected, non-commutative and every normal closed subgroup of $G$ ...
5
votes
0answers
103 views

Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$

Background There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. ...
2
votes
1answer
61 views

References about the exponent of automorphism groups of finite groups.

I will be pleased if one can help me to find any reference talking about the exponent of $\operatorname{Aut}(G)$, $G$ denotes a finite group. I'm specially interested to the case when $G$ is a finite ...
5
votes
0answers
82 views

Classes of groups known to be realizable (IGP)

A finite group $G$ of order $n$ is said to be realizable (over $\mathbb{Q}$) if there exists a Galois extension $L/\mathbb{Q}$ such that $\mathrm{Gal}(L/\mathbb{Q})=G$. I'm curious what classes of ...
4
votes
2answers
74 views

Reference Request for The Study of Abelian Groups

So I finished Lang's Algebra and after reading this partial Structure Theorem for abelian torsion groups that are not finitely generated , I've gotten interested in abelian groups, in particular ...
0
votes
0answers
45 views

Ring structure on finite string of elements of a group

This is a reference request. Suppose $(G, \cdot)$ is a group and consider the structure on $G^{<\omega}$ where, for $\mathbf{p} = (p_1, \dots, p_n) \in G^{<\omega}$ and $\mathbf{q} = (q_1, ...
6
votes
0answers
144 views

Historical Question about Schur-Zassenhaus Theorem

I couldn't find any historical information about Schur-Zassenhaus theorem in many books or even papers which mention this theorem. I think, Schur proved that if $G$ is a finite group and if $N$ is ...