All Questions

Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ? I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...

The following formula of astonishing beauty and power (imho): $$ \sum_{n \ge 0} \frac{| \mathrm{Hom}(G,S_n) | }{n! } z^n = \exp\left( \sum_{n \ge 1} \frac{|\text{Index}~n~\text{subgroups of}~ G|}nz^...

Let $n \in \mathbb{N}$. Is it true that for any $a, b, c \in \mathbb{N}$ satisfying $1 < a, b, c \leq n-2$ the symmetric group ${\rm S}_n$ has elements of order $a$ and $b$ whose product has order $...

Fix an algebraically closed field $K$ (maybe of characteristic zero first for simplicity, like $\mathbb{C}$). Given a partition $p=[a_1,...,a_m]$ of an integer $n$. We can identify $p$ with the ...

Let $\mathfrak{S}_n$ be the permutation group on an $n$-element set. For each fixed $k\in\mathbb{N}$, consider the two sets $$A_n(k)=\{\sigma\in\mathfrak{S}_n\vert\,\, \text{$\exists i,\,\, 1\leq i\...

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Let $f\in \Hom((\Bbb Z/p\Bbb Z)^2,\GL_{4}(\Bbb Z / p\Bbb Z))$ be an injective homomorphism. What is the number of ...