A characterization of C 2 ( q ) where q > 5

Ali Iranmanesh; Behrooz Khosravi

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 1, page 9-21
  • ISSN: 0010-2628

Abstract

top
The order of every finite group G can be expressed as a product of coprime positive integers m 1 , ⋯ , m t such that π ( m i ) is a connected component of the prime graph of G . The integers m 1 , ⋯ , m t are called the order components of G . Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups C 2 ( q ) where q > 5 are also uniquely determined by their order components. As corollaries of this result, the validities of a conjecture by J.G. Thompson and a conjecture by W. Shi and J. Be for C 2 ( q ) with q > 5 are obtained.

How to cite

top

Iranmanesh, Ali, and Khosravi, Behrooz. "A characterization of $C_2(q)$ where $q>5$." Commentationes Mathematicae Universitatis Carolinae 43.1 (2002): 9-21. <http://eudml.org/doc/248976>.

@article{Iranmanesh2002,
abstract = {The order of every finite group $G$ can be expressed as a product of coprime positive integers $m_1,\dots , m_t$ such that $\pi (m_i)$ is a connected component of the prime graph of $G$. The integers $m_1,\dots , m_t$ are called the order components of $G$. Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups $C_2(q)$ where $q>5$ are also uniquely determined by their order components. As corollaries of this result, the validities of a conjecture by J.G. Thompson and a conjecture by W. Shi and J. Be for $C_2(q)$ with $q>5$ are obtained.},
author = {Iranmanesh, Ali, Khosravi, Behrooz},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {prime graph; order component; finite group; simple group; prime graphs; order components; finite simple groups},
language = {eng},
number = {1},
pages = {9-21},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A characterization of $C_2(q)$ where $q>5$},
url = {http://eudml.org/doc/248976},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Iranmanesh, Ali
AU - Khosravi, Behrooz
TI - A characterization of $C_2(q)$ where $q>5$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 1
SP - 9
EP - 21
AB - The order of every finite group $G$ can be expressed as a product of coprime positive integers $m_1,\dots , m_t$ such that $\pi (m_i)$ is a connected component of the prime graph of $G$. The integers $m_1,\dots , m_t$ are called the order components of $G$. Some non-abelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symplectic groups $C_2(q)$ where $q>5$ are also uniquely determined by their order components. As corollaries of this result, the validities of a conjecture by J.G. Thompson and a conjecture by W. Shi and J. Be for $C_2(q)$ with $q>5$ are obtained.
LA - eng
KW - prime graph; order component; finite group; simple group; prime graphs; order components; finite simple groups
UR - http://eudml.org/doc/248976
ER -

References

top
  1. Chen G.Y., On Frobenius and 2 -Frobenius group, J. Southwest China Normal Univ. 20 (5) (1995), 485-487. (1995) 
  2. Chen G.Y., A new characterization of G 2 ( q ) , [ q ≡ 0 ( m o d 3 ) ] , J. Southwest China Normal Univ. (1996), 47-51. (1996) MR1374160
  3. Chen G.Y., A new characterization of sporadic simple groups, Algebra Colloq. 3 1 (1996), 49-58. (1996) Zbl0851.20011MR1374160
  4. Chen G.Y., On Thompson's conjecture, J. Algebra 185 (1996), 184-193. (1996) Zbl0861.20018MR1409982
  5. Chen G.Y., Further reflections on Thompson's conjecture, J. Algebra 218 (1999), 276-285. (1999) Zbl0931.20020MR1704687
  6. Chen G.Y., A new characterization of Suzuki-Ree groups, Sci. in China (Ser. A) 27 (5) (1997), 430-433. (1997) MR1481680
  7. Chen G.Y., A new characterization of E 8 ( q ) , J. Southwest China Normal Univ. 21 (3) (1996), 215-21. (1996) MR1374160
  8. Chen G.Y., A new characterization of P S L 2 ( q ) , Southeast Asian Bull. Math. 22 (1998), 257-263. (1998) MR1684163
  9. Gruenberg K.W., Roggenkamp K.W., Decomposition of the augmentation ideal and of the relation modules of a finite group, Proc. London Math. Soc. 31 (1975), 146-166. (1975) Zbl0313.20004MR0374247
  10. Iranmanesh A., Alavi S.H., A new characterization of A p where p and p - 2 are primes, Korean J. Comput. & Appl. Math. 8 3 (2001), 665-673. (2001) Zbl1013.20010MR1848926
  11. Iranmanesh A., Alavi S.H., Khosravi B., A characterization of P S L ( 3 , q ) where q is an odd prime power, J. Pure Appl. Algebra, to appear. MR1904845
  12. Iranmanesh A., Khosravi B., A characterization of F 4 ( q ) where q is even, Far East J. Math. Sci. 6 (2) (2000), 853-859. (2000) MR1808700
  13. Kondtrat'ev A.S., Prime graph components of finite groups, Math. USSR-sb. 67 (1) (1990), 235-247. (1990) MR1015040
  14. Shi W., A new characterization of the sporadic simple groups, Group Theory, Proceeding of the 1987 Singapore Group Theory Conference, Walter de Gruyter, Berlin, New York, 1989, pp. 531-540. Zbl0657.20017MR0981868
  15. Shi W., Pure quantitative characterization of finite simple groups I, Progress in Natural Science 4 (3) (1994), 316-326. (1994) MR1402664
  16. Shi W., Jianxing Bi, A new characterization of some simple groups of Lie type, Contemporary Math. 82 (1989), 171-180. (1989) MR0982286
  17. Shi W., Jianxing Bi, A characteristic property for each finite projective special linear group, Lecture Notes in Mathematics 1456 (1990), 171-180. Zbl0718.20009MR1092230
  18. Shi W., Jianxing Bi, A new characterization of the alternating groups, Southeast Asian Bull. Math. 17 (1) (1992), 81-90. (1992) Zbl0790.20030MR1173612
  19. Steinberg R., Automorphism of finite linear groups, Canad. J. Math. 12 (1960), 606-615. (1960) MR0121427
  20. Williams J.S., Prime graph components of finite groups, J. Algebra 69 (1981), 487-513. (1981) Zbl0471.20013MR0617092

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.