All Questions
130 questions
9
votes
5
answers
2k
views
A catalog of faithful representations of finite groups?
I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$.
Where ...
- 7,633
4
votes
1
answer
869
views
Finite Unipotent Groups: References
It will be a great pleasure for me if one can suggest "Survey Articles" on following topics related to the finite unipotent group $U(n,\mathbb{F}_q)$. (Thanks in advance!!!)
The number of ...
- 1,169
42
votes
6
answers
4k
views
Measures of non-abelian-ness
Let $G$ be a finite non-abelian group of $n$ elements.
I would like a measure that intuitively captures the
extent to which $G$ is non-commutative.
One easy measure is a count of the non-commutative ...
- 151k
3
votes
0
answers
209
views
What is known about 2-modular representations of Ree groups of type $F_4$?
A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...
- 52.9k
3
votes
0
answers
153
views
On divisors occurring as subgroup sizes
Given a finite group $G$ define $D(G)$ to be the number of divisors $r$ of $|G|$ for which there exists a subgroup of $G$ of order $r$.
Clearly $D(G) \leq d(|G|)$, where $d(n)$ denotes the number of ...
- 731
3
votes
0
answers
264
views
How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?
This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}...
- 1,173
4
votes
1
answer
686
views
Character theory of $2$-Frobenius groups.
This is a crosspost of my (slightly longer) question on MSE since I'm not getting any responses there.
Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\left(G/...
- 1,173
15
votes
4
answers
1k
views
Realizable Order Sequences for Finite Groups
My post is motivated at least in part by this MO question.
Has there been any work done on realizable order sequences for finite groups? By an "order sequence" I mean a non-decreasing list of the ...
- 7,831
0
votes
1
answer
227
views
What is a "non-splitting covering" of a finite group?
Apologies if this is elementary, but I have never heard the terminology before:
What is a "non-splitting covering" of a finite group?
I encountered the term while reading this paper, in which ...
- 1,173
16
votes
2
answers
992
views
Maximal number of maximal subgroups
Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ ...
- 2,179
13
votes
1
answer
1k
views
Convenient reference for subgroups of a finite semidirect product?
Given a finite group $G= H \ltimes N$ (with no particular constraints on $H, N$), it's probably been known for a long time how to describe efficiently the possible subgroups of $G$. A graduate ...
- 52.9k
11
votes
4
answers
2k
views
Textbook source for finite group properties deducible from character table?
Various questions have been posted on MO (some answered, some not) involving the character table of a finite group $G$ over a splitting field such as $\mathbb{C}$ of characteristic 0. My basic ...
- 52.9k
6
votes
2
answers
856
views
Algorithm for Brauer lifting via Brauer tree?
Background: Given a finite group $G$ and a prime $p$ dividing its order, Brauer theory compares the ordinary characters of $G$ with the Brauer characters arising from $p$-modular representations. On ...
- 52.9k
5
votes
2
answers
281
views
Doubly covering an even lattice
I have read that there is a way to construct a group which is a double cover of an even lattice. The very tantalizing thing about this is that if the even lattice is chosen to be the Leech lattice, ...
- 3,645
9
votes
2
answers
485
views
Reference for restriction of a simple module over a splitting field to a smaller field?
This is mainly a request for a straightforward reference (preferably at textbook level). The question comes up while responding to a question raised by non-specialists in finite group representations....
- 52.9k
7
votes
2
answers
571
views
abelian centralizers in almost simple groups
Hallo!
I'm looking for a reference. I'm sure that the information I need is already in the literature but I'm having some trouble to find it. Here is the question.
Let $S$ be a non-abelian finite ...
- 73
8
votes
1
answer
446
views
Radical of $F_p[SL(2,p)]$
Let $G=SL(2,p)$. Does anyone know what is the radical of the group algebra $F_p[G]$?
Does there exists any book/paper where it is calculated?
By radical here I mean maximal ideal I of $F_p[G]$ such ...
- 2,179
7
votes
0
answers
430
views
The maximal order of an element in orthogonal groups over finite fields of characteristic 2
Let $q$ be a power of $2$ and let $(V,Q)$ be a
quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type ...
- 7,609
32
votes
0
answers
993
views
Is there a Mathieu groupoid M_31?
I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
- 3,645
11
votes
2
answers
4k
views
Orders of automorphism groups of p-groups
There is a theorem that says that if $p$ is a prime and $G$ is a $p$-group with $|G| = p^{n}$, $|Aut(G)|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$.
This theorem is sharp, since $\Pi_{k=0}^{n-1} (p^{n}-...
- 3,645
35
votes
6
answers
5k
views
Character-free proof that Frobenius kernel is a normal subgroup?
The question is in the title, but here is some background/reminders:
A subgroup $H\neq\{1\}$ of a finite group $G$ is called a Frobenius complement if $H\cap H^g = \{1\}$ for all $g\in G\backslash H$....
- 13k
5
votes
1
answer
264
views
Group not leaving subset invariant
Let $Y,X$ be two sets of size n,m. Let $Y\subset X$.
What is the maximal group(in size) $G< Sym(X)$ such that gY=Y imply that $g=1$?
Here I mean that the only permutation which permutes elements of ...
- 2,179
12
votes
4
answers
2k
views
Efficient presentations for finite groups
A finitely presented group which has more generators than relations has an infinite abelianization and so is an infinite group. Therefore, for a finite group, all presentations must have at least as ...
- 3,645
7
votes
1
answer
958
views
Groups whose normal subgroups form a chain with respect to inclusion
Let G be a finite group. In general, given two normal subgroups N and K of G, we need not have N < K or K < N. The easiest example is the Klein 4-group V4 and its subgroups of order 2. So assume ...
- 307
3
votes
1
answer
696
views
Unique factorization of finite groups under direct sum?
I am told that finite groups have unique factorization under direct product. That is, call a nontrivial group "indivisible" if it is not isomorphic to a direct product of nontrivial groups. Then ...
- 4,994
17
votes
3
answers
1k
views
How to find more (finite almost simple) groups with a given Sylow subgroup
I'm looking for some examples of actions on Sylow p-groups, and often those actions appear in the case of finite almost simple groups. Given a finite almost simple group, I understand in principle ...
- 10.7k
7
votes
2
answers
780
views
Finite groups with a character having very few nonzero values?
A number theorist I know (who studies Galois representations) raised a question recently:
Which finite groups can have an irreducible character of degree at least 2 having only $n=2, 3$, or 4 ...
- 52.9k
5
votes
2
answers
346
views
Reference request: A theorem by S. Garrison
A theorem by S. Garrison states that if $G$ is a finite solvable group and $|cd(G)| = 4$ then $dl(G)\leq |cd(G)|$ (the Taketa inequality, which is conjectured to hold for all finite solvable groups). ...
- 2,508
44
votes
10
answers
11k
views
The finite subgroups of SL(2,C)
Books can be written about the finite subgroups of $\mathrm{SL}(2,\mathbb C)$ (and their immediate family, like the polyhedral groups...) I am about to start writing notes for a short course about ...
- 47.3k
17
votes
5
answers
3k
views
Reference for this theorem in representation theory?
Let $G$ be a finite group and $\chi$ be an irreducible character of
$G$ (characteristic zero algebraically closed base field). If $H$ is
the kernel of $\chi$ then the irreducible representations of $G/...
- 887