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 ...
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 ...
0
votes
1answer
70 views

Character Table of the Monster

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

the best book for exercising finite groups theory. [duplicate]

I need a book in finite group theory which contain lots of question and also with answers,the answers is important for me because I need to compare my answers to the the right answers and learn right ...
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 $S_4$ ...
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
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" ...
6
votes
1answer
78 views

Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been ...
5
votes
3answers
185 views

Symmetries of combinatorial structures.

Studying the automorphism groups of graphs/finite geometries/designs has been quite useful and important for both group theory and combinatorics. I know of the following books which cover the ideas ...
2
votes
1answer
151 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 ...
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 ...
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 ...
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 ...
1
vote
1answer
94 views

Classification of group extensions

For hours I have been looking for " Claude Archer. Classification of group extensions. PhD thesis,Université Libre de Bruxelles, 2002 " but I found nothing . Is there any replacement for this thesis , ...
3
votes
1answer
138 views

Generalizations of fitting subgroup

The Fitting subgroup of a group $G$ has two generalizations: the generalized Fitting subgroup $F^*(G)$ of Bender and $\tilde F(G)$ of Schmid. The latter is defined by $\tilde F(G)/\Phi(G) = ...
-3
votes
1answer
102 views

Books about finite groups construction [duplicate]

The algorithms to construct all finite groups of a given order introduced in : A millennium project: constructing small groups, Hans Ulrich Besche, Bettina Eick, E.A. O'Brien, Internat. J. Algebra ...
16
votes
1answer
336 views

Books to understand the construction of all groups of a specific order

The algorithms introduced by Besche–Eick (1999) were used to construct (or count) the groups of order up to 2000 in Besche–Eick–O'Brien (2002), yet I find the algorithms somewhat inaccessible. How ...
6
votes
2answers
172 views

Elementary Abelian $p$-Subgroups of $GL_n(\mathbb{F}_p)$

Let $p$ be a prime number. If $G$ is a finite group, an elementary abelian $p$-subgroup of $G$ of rank $r$ is any group $E\subset G$ such that $E\cong (\mathbb{Z}/p)^{\oplus r}.$ Let $\mathcal{E}_r$ ...
1
vote
2answers
50 views

Number of abelian groups Vs Number of non-abelian groups

I would like to see a table that shows the number of non-abelian group for every order n. It is a preferable if the table contains the number of abelian groups of order n (this is not necessary ...
0
votes
4answers
131 views

For which finite groups, number of elements of order $p$ is not $p-1$?

Let $G$ be a finite group and $p\mid |G|$ be prime. Can $G$ have exactly $p-1$ elements of order $p$? (except trivial groups which are isomorphic to $\Bbb Z_p$) I remember something similar to it. ...
3
votes
1answer
62 views

How are these infinite classes of groups of orders $6n$ and $8n$ called?

In the book Gordon James, Martin Liebeck: Representations and Characters of Groups the following three classes of groups are given in a series of exercises, where the reader is asked to find all their ...
2
votes
0answers
44 views

characteristic-$p$-type groups, and the Borel-Tits theorem for $PSL(V)$

If $G$ is a finite group and $F^*(G)$ is the generalized Fitting subgroup we say that $G$ has characteristic $p$ ($p$ is a prime that divides $|G|$) if $$F^*(G)=O_p(G)$$ Moreover $G$ is a ...
1
vote
1answer
104 views

Definition of induced representation by tensor product

Suppose there is a finite group $G$ with a subgroup $H$ an some field $K$. If one has a representation of the subgroup, one can construct the induced representation $\rho:=Ind_H^G$ according to ...
5
votes
1answer
183 views

Sylow $p$-Subgroups of Classical Groups over $\mathbb{F}_p$

Let $p$ be a prime, and let $G$ be any of the finite classical groups $SL_n(\mathbb{F}_p)$, $O_n(\mathbb{F}_p)$, or $SP_n(\mathbb{F}_p)$. Let $P$ be a Sylow $p$-subgroup of $G$. What is $P$ as a ...
7
votes
2answers
141 views

Do group rings appear outside of representation theory?

I am particularly concerned with finite groups. I have seen group rings used in the fundamentals of representation theory as the dual notion to representations. I haven't ever seen them anywhere ...
4
votes
1answer
176 views

Irreducible subgroups of $\text{GL}(2,p)$

I just wonder any sources that provides the list of all minimal irreducible subgroups of $\text{GL}(2,p)$. Here the term "minimal" means there is no proper subgroup that is irreducible. A list of ...
13
votes
1answer
152 views

The classification all finite groups which possess a single proper non-trivial normal subgroup

We know that For $n≥5$, $A_n$ is the only proper nontrivial normal subgroup of $S_n$. I am kindly asking to know the possible presented references including the following point, if anybody is ...
3
votes
3answers
103 views

Does there exist a finite group with the following presentation?

Let $G$ be a finite group (with only two generators and $m=n$) presented as $$ G = \langle a, b : a^m = b^n = (W(a,b))^p= \ldots\text{other-such-relations}\ldots= 1 \rangle $$ where $m,n,p>1$ , ...
12
votes
0answers
128 views

References on the theory of $2$-groups.

Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
2
votes
1answer
75 views

Details about “fingerprinting” algorithms for groups?

where can I find details about "Fingerprinting" algorithms (to test whether two groups are non-isomorphic) "‘Fingerprinting’: For every group $G_1,…, G_r$ evaluate various isomorphism-invariant ...
3
votes
2answers
123 views

Sequence and series problems with slick group theoretic solutions.

So I'm trying to get this unmotivated student in my class interested in learning group theory. I've recently ascertained that he likes analysis a bit better than algebra, and that sequences and ...
11
votes
3answers
687 views

Choosing an advanced group theory text: concerns

In this question, An Introduction to the Theory of Groups by Rotman is recommended twice as a good second-course group theory text. However, after reading the reviews here, and seeing this pdf of ...
4
votes
1answer
66 views

Induced representations and degrees

Let $G$ be a finite group, $S$ a subgroup, $K$ an arbitrary (not necessarily algebraically closed) field whose characteristic does not divide the group order. 1) Let $\operatorname{Ind}_S^G(f)$ have ...
2
votes
1answer
71 views

Burnside's Action Equivalence Criterion

Theorem: Actions of a finite group G on finite sets $X$ and $Y$ are equivalent if and only if $|X^H|=|Y^H|$ for each subgroup $H$ of $G$ I've proved the finite case inductively but I'm wondering ...
3
votes
0answers
78 views

References about finite group theory

In your opinion which are the best books regarding the theory of finite groups? I think that a wonderful one is "Finite Group Theory - Michael Aschbacher". Many thanks.
1
vote
1answer
96 views

Infinite index subgroups of $SL(3,\mathbf{Z})$

Finite index subgroups of $SL(3,\mathbf{Z})$ are well-know (at least we know that they are congruence subgroups). But I wasn't able to find reference on infinite index subgroups. Does someone knows ...
6
votes
2answers
286 views

Involutions and Abelian Groups, II.

In the thread Involutions and Abelian Groups, I supplied a solution to the following interesting problem (with the help of hints provided by the OP). Let $ G $ be a finite group and $ I(G) $ the ...
23
votes
1answer
321 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
8
votes
1answer
337 views

Is there a counterexample to this weakened converse of Hall's theorem?

Suppose that a finite group $G$ contains a Hall $\{p,q\}$-subgroup for every pair of prime divisors $p,q$ of $|G|$. Does it follow that $G$ is solvable?
5
votes
2answers
259 views

Looking for texts in representation theory

I recently finished a course in representation theory, and while I learned a lot from it, I know that there's a lot more in the subject that I missed. For the course we used Fulton and Harris as a ...
2
votes
1answer
224 views

Sum of squares of dimensions of irreducible characters.

For anyone familiar with Artin's Algebra book, I just worked through the proof of the following theorem, which can be seen here: (5.9) Theorem Let $G$ be a group of order $N$, let ...
2
votes
2answers
209 views

Where can I find the original papers by Frobenius concerning solutions to $x^n = 1$ in a finite group?

A theorem proven by Frobenius states that If $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Articles discussing this theorem ...
1
vote
2answers
220 views

Number of prime divisors of element orders from character table.

From wikipedia: It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group ...
4
votes
2answers
142 views

Gentle introduction into stability and classification theory

I am badly looking for a (very) gentle introduction into stability and classification theory answering at least some of the following questions: Why is a stable theory called "stable"? What is a ...
3
votes
3answers
147 views

Constructing a basis for finite abelian groups

Let $G$ be a finite abelian group, and $g_1, \ldots, g_k$ a set of "linearly independent elements", namely such that $\langle g_1 \rangle \oplus \ldots \oplus\langle g_k \rangle$. I would like to ...
3
votes
1answer
153 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
3
votes
3answers
76 views

How to expand a representation

If there is a finite group with a normal subgroup and a representation of this subgroup over a finite field. How can one expand this representation to a representation of the whole group? Are there ...
6
votes
0answers
287 views

Resource: Group Theory

There is a website providing recent thesis in finite geometry. Is there any website with collection of recent thesis on (finite) group theory. I want to see the Ph.D. thesis of Raymond T. Shepherd, ...
2
votes
0answers
49 views

Subgroups of semi-direct products arising from fixed-point-free actions

I am interested in subgroups of semidirect products arising from fixed-points-free actions. Suppose you have a group $A$ acting fixed-point-freely on a group $N$. Can we describe completely the ...
1
vote
1answer
52 views

Inconjugate maximal subgroups of a soluble group

I am looking for a proof of the following proposition: If $M_1,M_2$ are inconjugate maximal subgroups of the finite and soluble group $G$, then $M_1\cap M_2$ is maximal in at least one of $M_1$ or ...