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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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?

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 …

Let $G$ be a finite group and $H$ be a nilpotent subgroup of $Aut(G)$. If $C_{G}(H)=1$, is $G$ solvable?

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 …

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 …

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 …

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 …