Examples for embeddings of groups \(\text{PSL}_ 2(F_ p)\) into simple Lie groups
The main theorem is as follows: Let \(G\) be a connected simple linear algebraic group over an algebraically closed field \(k\) with Coxeter number \(h\). Then (1) If \(p=h+1\), then \(G(k)\) contains a subgroup isomorphic to \(\text{PGL}_2(\mathbb{F}_p)\), except when \(\text{char}(k)=2\), \(G\cong\text{PGL}_2\); (2) If \(p=2h+1\), then \(G(k)\) contains a subgroup isomorphic to \(\text{PSL}_2(\mathbb{F}_p)\). To prove this theorem, first under the assumption that \(\text{char}(k)=p\geq h\) the author constructs a ``principal'' embedding \(\text{PGL}_2(\mathbb{F}_p)\to G(\mathbb{F}_p)\). If \(p=h+1\), then the above embedding can be lifted to an embedding \(\text{PGL}_2(\mathbb{F}_p)\to G(\mathbb{Z}_p)\) where \(\mathbb{Z}_p\) is the ring of \(p\)-adic integers. If \(p=2h+1\), then it can be lifted to an embedding \(\text{PSL}_2(\mathbb{F}_p)\to G(\mathbb{Z}_p[\sqrt{\pm p}])\). Finally the Bruhat-Tits theory is used to complete the base change from characteristic zero to arbitrary characteristic.
connected simple linear algebraic groups
scientific article; zbMATH DE number 866101