Abstract
Let \(r\) be a prime and \(G\) be a finite group, and let \(R, \,S\) be Sylow \(r\)-subgroups of \(G\) and \(\text{ PGL }(2, r)\) respectively. We prove the following results: (1) If \(|G|=|\text{ PGL }(2, r)|\) and \(|N_{G}(R)|=|N_{\mathrm{PGL}(2, r)} (S)|\) and \(r\) is not a Mersenne prime, then \(G\) is isomorphic to \(\text{ PSL } (2, r) \times C_{2}, \,\text{ SL }(2, r)\) or \(\text{ PGL }(2, r)\). (2) If \(|G|=|\text{ PGL }(2, r)|, \,|N_{G}(R)|=|N_{\mathrm{PGL}(2, r)}(S)|\) where \(r>3\) is a Mersenne prime and \(r\) is an isolated vertex of the prime graph of \(G\), then \(G\cong \text{ PGL }(2, r)\).
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The author is thankful to the referee for carefully reading the paper and for his suggestions and remarks.
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Communicated by J. S. Wilson.
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Asboei, A.K. Characterization of projective linear groups by the order of the normalizer of a Sylow subgroup. Monatsh Math 173, 309–314 (2014). https://doi.org/10.1007/s00605-013-0493-2
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DOI: https://doi.org/10.1007/s00605-013-0493-2