On the prime graph of \(\text{PSL}(2,p)\) where \(p>3\) is a prime number.
The prime graph of a group \(G\) is a graph whose vertices are the primes which divide \(|G|\) and \(pq\) is an edge if there is an element \(x\in G\), \(o(x)=pq\). Let \(G=\text{PSL}(2,p)\), \(p\) a prime \(p\equiv 3\pmod 4\) and \(S\) an arbitrary finite group. Assume that \(G\) and \(S\) have isomorphic prime graphs, then the authors show that either \(S=\text{PSL}(2,p)\) or \(p=7\) or 11 and \(G/O_2(G)\cong\text{PSL}(2,8)\) in the first case or \(M_{11}\) in the second. The proof as usual in this area goes via the Kegel-Gruenberg result, which then basically reduces the proof to simple groups \(S\) and then using the classification of the finite simple groups and a case by case analysis gives then the result. At the end of the paper a similar result for \(p\equiv 1\pmod 4\) is stated and a sketch of a proof is given.
scientific article; zbMATH DE number 5355806
scientific article