2
votes
1answer
47 views

The comultiplication on $\mathbb{C} S_3$ for a matrix basis?

Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra. The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$. For $G=S_3$, ...
2
votes
1answer
47 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
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 ...
0
votes
0answers
18 views

Frobenius subgroups of $GL(n,2)$

I need to know the structure of Frobenius subgroups of $GL(n,2)$, where $n<14$. Is there any good reference? Many thanks.
2
votes
2answers
73 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 ...
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: ...
2
votes
0answers
49 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 ...
3
votes
2answers
101 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 ...
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 ...
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 ...
0
votes
1answer
114 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
23 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
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 $S_4$ ...
7
votes
1answer
116 views

Question about inverse Galois problem

I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over ...
1
vote
5answers
121 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
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" ...
6
votes
1answer
134 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
207 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 ...
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 ...
8
votes
1answer
94 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
97 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
177 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
101 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
153 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
112 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
387 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
204 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
52 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
137 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
66 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
49 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
126 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
214 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
156 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
211 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
158 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
109 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 $m,n,p>1$ , ...
14
votes
0answers
141 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
78 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
127 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
839 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
68 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
75 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
83 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
103 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
317 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 ...
25
votes
1answer
344 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 ...
9
votes
1answer
351 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
3answers
328 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
292 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 ...