Characterization of projective general linear groups

Document Type : Research Paper

Author

Farhangian University, Shariati Mazandaran, Iran

Abstract

‎Let $G$ be a finite group and $\pi_{e}(G)$ be the set of element orders of $G $‎. ‎Let $k \in \pi_{e}(G)$ and $s_{k}$ be the number of elements of order $ ‎k $ in $G$‎. ‎Set nse($G$):=$\{ s_{k} | k \in \pi_{e}(G)\}$‎. ‎In this paper‎, ‎it‎ ‎is proved if $|G|=|$ PGL$_{2}(q)|$‎, ‎where $q$ is odd prime power and nse$ ‎(G)= $nse$($PGL$_{2}(q))$‎, ‎then $G \cong $PGL$_{2}(q)$‎.

Keywords

Main Subjects


A‎. ‎K‎. ‎Asboei (2013). ‎A new characterization of PGL$_{2}(p)$. J‎. ‎Algebra Appl.. 12 (7), 0 A‎. ‎K‎. ‎Asboei and A‎. ‎Iranmanesh ‎A characterization of Linear group $L_{2}(p)$. Czechoslovak Math‎. ‎J., ‎(to appear). A‎. ‎K‎. ‎Asboei‎, ‎S‎. ‎S‎. ‎Amiri‎, ‎A‎. ‎Iranmanesh and A‎. ‎Tehranian (2012). ‎A characterization of Symmetric group $S_{r}$‎, ‎where $r$ is prime number. Ann‎. ‎Math‎. ‎Inform.. 40, 13-23 A‎. ‎K‎. ‎Asboei‎, ‎S‎. ‎S‎. ‎Amiri‎, ‎A‎. ‎Iranmanesh and A‎. ‎Tehranian (2013). ‎A new characterization of sporadic simple groups by NSE and order‎. J‎. ‎Algebra Appl.. 12 (2) A‎. ‎K‎. ‎Asboei‎, ‎S‎. ‎S‎. ‎Amiri‎, ‎A‎. ‎Iranmanesh and A‎. ‎Tehranian (2013). ‎A new characterization of $A_{7}$ and $A_{8}$. An‎. ‎Stiint‎. ‎Univ‎. ‎Ovidius Constanta Ser‎. ‎Mat.. 21 (3), 43-50 A‎. ‎R‎. ‎Moghaddamfar and W‎. ‎J‎. ‎Shi (2006). ‎The number of finite groups whose element orders is given. Beitrage Algebra Geom.. 47 (2), 463-478 A‎. ‎S‎. ‎Kondratev (1989). ‎On prime graph components of finite simple groups. Math‎. ‎Sb.. 180 (6), 787-797 A‎. ‎S‎. ‎Kondratev and V‎. ‎D‎. ‎Mazurov (2000). ‎Recognition of alternating groups of prime degree from the orders of their elements‎. Sib‎. ‎Math‎. ‎J.. 41 (2), 294-302 C‎. ‎G‎. ‎Shao‎, ‎W‎. ‎Shi and Q‎. ‎H‎. ‎Jiang (2008). ‎Characterization of simple $K_{4}-$groups. Front‎. ‎Math‎. ‎China.. 3, 355-370 D‎. ‎Gorenstein (1980). Finite Groups. ‎Chelsea Publishing Co.‎, ‎New York. D‎. ‎Passman (1968). Permutation Groups. ‎W‎. ‎A‎. ‎Benjamin‎, ‎Inc.‎, ‎New York-Amsterdam. G‎. ‎Frobenius (1895). ‎Verallgemeinerung des sylowschen satze. Berliner sitz.. , 981-993 G‎. ‎Y‎. ‎Chen (1995). ‎On structure of Frobenius and 2-Frobenius group. J‎. ‎Southwest China Normal University, (Chinese). 20 (5), 485-487 G‎. ‎Y‎. ‎Chen (1996). ‎On Thompson's conjecture. J‎. ‎Algebra. 185 (1), 184-193 J‎. ‎G‎. ‎Thompson (1959). ‎Normal $p$-complement for finite groups. Math‎. ‎Z.. 72, 332-354 J‎. ‎H‎. ‎Conway‎, ‎R‎. ‎T‎. ‎Curtis‎, ‎S‎. ‎P‎. ‎Norton and R‎. ‎A‎. ‎Wilson‎ (1985). Atlas of finite groups. ‎Oxford University Press‎, ‎Eynsham. J‎. ‎S‎. ‎Williams (1981). ‎Prime graph components of finite groups. J‎. ‎Algebra. 69 (2), 487-513 L‎. ‎Weisner (1925). ‎On the number of elements of a group which have a power in a given conjugate set. Bull‎. ‎Amer‎. ‎Math‎. ‎Soc.. 31, 492-496 N‎. ‎Iiyori and H‎. ‎Yamaki (1993). ‎Prime graph components of the simple groups of Lie type over the field of even characteristic. J‎. ‎Algebra. 155 (2), 335-343 V‎. ‎D‎. ‎Mazurov and E‎. ‎I‎. ‎Khukhro (1995). Unsolved Problems in group theory‎, ‎the Kourovka Notebook. ‎13 ed‎, ‎Russian Academy of Sciences Siberian Division‎, ‎Institute of Mathematics‎, ‎Novosibirsk. W‎. ‎Shi (1989). A new characterization of the sporadic simple groups in group theory. ‎In Proceeding of the 1987 Singapore Group Theory‎, ‎Conference‎, ‎Walter de Gruyter‎, ‎Berlin. , 531-540