Masoumeh Sajjadi
On the recognizability of some projective general linear groups by the prime graph

Comment.Math.Univ.Carolin. 63,4 (2022) 443-458.

Abstract: Let G be a finite group. The prime graph of G is a simple graph \Gamma(G) whose vertex set is \pi(G) and two distinct vertices p and q are joined by an edge if and only if G has an element of order pq. A group G is called k -recognizable by prime graph if there exist exactly k nonisomorphic groups H satisfying the condition \Gamma(G) = \Gamma(H). A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that {\rm PGL}(2,p^\alpha) is recognizable, if p is an odd prime and \alpha > 1 is odd. But for even \alpha , only the recognizability of the groups {\rm PGL}(2, 5^2), {\rm PGL}(2, 3^2) and {\rm PGL}(2, 3^4) was investigated. In this paper, we put \alpha = 2 and we classify the finite groups G that have the same prime graph as \Gamma({\rm PGL}(2, p^2)) for p=7, 11, 13 and 17. As a result, we show that {\rm PGL}(2, 7^2) is unrecognizable; and {\rm PGL}(2, 13^2) and {\rm PGL}(2, 17^2) are recognizable by prime graph.

Keywords: projective general linear group; prime graph; recognition

DOI: DOI 10.14712/1213-7243.2023.009
AMS Subject Classification: 20D05 20D60 20D08