All Questions

Is there a bound $B$ such that every 2-generator subgroup $G = \langle a, b \rangle \le {\rm GL}(2,\mathbb{Z})$ whose generators do not satisfy a relation of length $\leq B$ is free? If it exists, ...

Denote the set of $n\times n$ permutation matrices by $\mathfrak{S}_n$. The ordinary transpose preserves this group. Given $P\in\mathfrak{S}_n$, construct the $n\times n$ matrix ${}^tP$ according to ...

Let $V$ be a vector space. A reflection is a linear map $f: V \to V$ which has an eigenvalue $1$ with multiplicity $n-1$. Let $S_n$ be the symmetric group on $\{1,\ldots,n\}$. Then the reflections in ...