Some New Characterizations for PGL(2, q)

Abstract

Many authors introduced some characterizations for finite groups. In this paper as the main result we prove that the finite group PGL(2, q) is uniquely determined by its noncommuting graph. Also we prove that PGL(2, q) is characterizable by its noncyclic graph. Throughout the proof of these results we prove that PGL(2, q) is uniquely determined by its order components and using this fact we give positive answer to a conjecture of Thompson and another conjecture of Shi and Bi for the group PGL(2, q). RESUMENMuchos autores introdujeron algunas caracterizaciones de los grupos finitos. En este trabajo como principal resultado se demuestra que grupo finito PGL(2, q) es determinado nicamente por su gráfica no conmutativa. También se demuestra que PGL(2, q) 13, 2 (2011) es caracterizable por su gráfico no cíclico. A lo largo de la prueba de estos resultados se demuestra que PGL(2, Q) es determinadoúnicamente por los componentes de su orden y con ello damos respuesta positiva a una conjetura de Thompson y otra conjetura de Shi Bi y para el grupo PGL(2, q).

PGL2(q) cannot be determined by its cs

“…In [14] and [17], it has been shown that Thompson's conjecture is true for some almost simple groups. Inspired by Thompson's conjecture, A. Camina and R. Camina come up with the following problem [10]: Problem.…”