NCF-distinguishablity by prime graph of \(\text{PGL}(2,p)\) where \(p\) is a prime
From the authors' summary: Let \(G\) be a finite group. The prime graph \(\Gamma(G)\) of \(G\) is defined as follows. The vertices of \(\Gamma(G)\) are the primes dividing the order of \(G\) and two distinct vertices \(p\), \(p'\) are joined by an edge if there is an element in \(G\) of order \(pp'\). Let \(p\) be a prime number. We know that \(\text{PGL}(2,p)\) has a unique nonabelian composition factor which is isomorphic to \(\text{PSL}(2,p)\). Assume that \(p\) is not a Mersenne or Fermat prime and \(p\neq 11\), \(19\). In this paper, we determine the structure of finite groups with the same prime graph as \(\text{PGL}(2,p)\) and prove that if \(G\) is a finite group such that \(\Gamma(G)=\Gamma(\text{PGL}(2,p))\) and \(p\neq 13\), then \(G\) has a unique nonabelian composition factor which is isomorphic to \(\text{PSL}(2,p)\) and if \(p=13\), then \(G\) has a unique nonabelian compostion factor which is isomorphic to \(\text{PSL}(2,13)\) or \(\text{PSL}(2,27)\).
scientific article; zbMATH DE number 5959796
scientific article