3
votes
2answers
121 views
On finite groups with same complex-valued character table
What are the necessary and sufficient conditions for two finite groups $G$ and $H$
to have same complex-valued character table?
Is there any criterion for which one could know abou …
21
votes
1answer
463 views
Why are Schur multipliers of finite simple groups so small?
Given a finite simple group $G$, we can consider the quasisimple extensions $\tilde G$ of $G$, that is to say central extensions which remain perfect. Some basic group cohomology …
4
votes
1answer
132 views
Good effective versions of theorems of Artin and Brauer
The theorem of Artin and Brauer of the title are the famous theorem in the theory of representation of finite groups.
For example, Artin's theorem is the statement that for every …
3
votes
4answers
359 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 $\ma …
0
votes
0answers
71 views
How to find quotients of infinite triangle groups or von Dyck groups?
I need the following information about the quotients of infinite triangle (or von Dyck) groups.
(1) Let $G(l,m,n)$ defined as $S^l$ =$T^m$ = $(ST)^n$ = $E$ is the hyperbolic ($1/l …
3
votes
1answer
85 views
Special automorphisms of extraspecial groups
Let $G$ be an extraspecial group of order $p^{2r+1}$ (where $p$ is an odd prime), and let $V$ be a faithful representation of $G$. Consider the normal subgroup $H$ of $Aut(G)$ cons …
0
votes
1answer
74 views
Odd-order groups with homocyclic sylow subgroups
We say that $G$ is a homocyclic group, if it is direct product of isomorphic cyclic groups.
Is there any classification of finite odd-order groups which all their Sylow subgroups a …
3
votes
2answers
244 views
group generated by Coxeter elements
Let $G$ a connected semisimple simply connected group over $\mathbb{C}$ and $W$ his Weyl group.
What can be said about $W'$, the subgroup of $W$ generated by the Coxeter elements …
3
votes
1answer
111 views
Relationship between the number of Sylow subgroups with element orders in finite group
In a finite group what is relationship between the number of Sylow $p$-subgroups with the number of elements of order a multiple of $p$?
Is there any reference for my question?
9
votes
2answers
476 views
Embeddings of finite groups into GL(n,Q_p)
This question is inspired by some interesting comments on this recent question.
Fix an integer $n \geq 1$ and a finite subgroup $G$ of $\mathrm{GL}_n(\mathbf{C})$. It is known tha …
4
votes
0answers
181 views
nilpotent fixed-point-free groups of automorphisms
Let $G$ be a finite group and $H$ be a nilpotent subgroup of
$Aut(G)$. If $C_{G}(H)=1$, is $G$ solvable?
3
votes
3answers
217 views
Intersection of all normalizers
This is probably standard for group-theorists:
Let $G$ be a finite group. Is it true that the intersection of all normalizers of subgroups equals the center?
If so, where do I find …
17
votes
6answers
1k 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 t …
10
votes
2answers
312 views
Finite subgroups of $PGL(3,K)$
It is well-known that finite subgroups of $PGL_2(\mathbb{C})$ are cyclic groups, dihedral groups, A4, S4 and A5 and each of these groups occurs exactly once (up to conjugacy). Thes …
0
votes
0answers
52 views
Computing the number of elements of order $2$ and $3$, in the groups $L_{3}(q)$
What are the number of elements of order $2$ and $3$ in the groups $L_{3}(q)$?
Also let $r$ be a divisor of $q^{2}+q+1$. What is the number of elements of
order $r$ in the groups …

