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

[摘要] 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 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.

[学科分类] 物理化学和理论化学

[关键词] projective general linear group;prime graph;recognition