Some new characterizations for PGL(2, q)

B. Khosravi; M. Khatami; Z. Akhlaghi

doi:10.4067/S0719-06462011000200009

Authors

B. Khosravi School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran.
M. Khatami Dept. of Pure Math., Faculty of Math. and Computer Sci., Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, Iran.
Z. Akhlaghi Dept. of Pure Math., Faculty of Math. and Computer Sci., Amirkabir University of Technology (Tehran Polytechnic), 424, Hafez Ave., Tehran 15914, Iran.

DOI:

https://doi.org/10.4067/S0719-06462011000200009

Keywords:

Noncommuting graph, prime graph, noncyclic graph, order components

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).