Tagged Questions

0
votes
4answers
100 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
55 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
30 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
27 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 ...
3
votes
1answer
64 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 ...
6
votes
2answers
103 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 ...
3
votes
1answer
92 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 ...
12
votes
1answer
104 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
62 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 $m,n,p>1$ , ...
11
votes
0answers
80 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
62 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
86 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 ...
9
votes
2answers
280 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 ...
3
votes
0answers
52 views

Induced representation and degrees

Let $G$ be a finite group, $S$ a subgroup, $K$ an arbitrary (not necessarily algebraically closed) field which characteristic does not divide the group order. 1) Let $Ind_S^G(f)$ have the degree $n$ ...
1
vote
1answer
56 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
55 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
61 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
131 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 ...
22
votes
1answer
252 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 ...
7
votes
1answer
302 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?
4
votes
2answers
136 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
67 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
124 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
128 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 ...
0
votes
0answers
27 views

Irreducible $2$-dimensional $\mathbb{C}$-representation of $D_{16}\times C_{2}$.

I am looking for all irreducible $2$-dimensional $\mathbb{C}$-representation of $D_{16}\times C_{2}$, where $D_{16}$ is the dihedral group of order $16=8\times 2$ and $C_{2}$ is the cyclic group of ...
4
votes
2answers
105 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
99 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
139 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
70 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
246 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
44 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
42 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 ...
4
votes
1answer
120 views

Thompson's Conjecture

I have heard that the following is a conjecture due to Thompson: The number of maximal subgroups of a (finite) group $G$ does not exceed the order $|G|$ of the group. My question is: did Thompson ...
4
votes
1answer
99 views

Is there an algorithm to determine whether rational matrices generate a finite group?

This is inspired by this question. Given finitely many invertible rational $n\times n$ matrices $A_{1},\ldots, A_{k}\in\operatorname{GL}(n,\mathbb{Q})$, is there an algorithm (a practical one) to ...
6
votes
1answer
145 views

Reference for classification of small groups

There are various online resources for the classification of groups of small order, such as this one or that one. Is there any nice reference in the literature which contains such a classification ...
5
votes
2answers
184 views

Reference for the subgroup structure of $PSL(2,q)$

This material is covered in detail in Dickson's "Linear Groups with an exposition of the Galois Field Theory", chapter XXII and Huppert's "Endliche Gruppen", chapter II, paragraph 8. Since I don't ...
7
votes
3answers
538 views

Three finite groups with the same numbers of elements of each order

There exist pairs of finite groups $G$ and $H$ such that $G$ and $H$ are not isomorphic, yet they have the same number of elements of each order. For example, if $p$ is an odd prime, then the group ...
1
vote
1answer
234 views

Induction from normal subgroups

Let $G$ be a (finite) group and $N$ a normal subgroup. Given an irreducible representation $\pi$, how can I decompose $Ind_N^G \pi$? I'd be happy also about a good reference for this.
4
votes
8answers
460 views

Order of cyclic groups

Wikipedia says: It is known that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if n is 1 or 2 or 4 or $p^k$ or $2p^k$ for an odd prime number p and k ≥ 1. The statement seems provable ...
3
votes
1answer
328 views

Exercises in Group Cohomology

I'm interested in finding a textbook to learn group cohomology, a book that contains a lot of examples and also a lot of good exercises to test my understanding. I would appreciate some feedback. ...