Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON NON-ISOMORPHIC GROUPS WITH THE SAME SET OF ORDER COMPONENTS

  • Darafsheh, Mohammad Reza (Department of Mathematics, Statistics and Computer Science Faculty of Science, University of Tehran)
  • Published : 2008.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we will prove that the simple groups $B_p(3)\;and\;G_p(3)$, p an odd prime number, are 2-recognizable by the set of their order components. More precisely we will prove that if G is a finite group and OC(G) denotes the set of order components of G, then OC(G) = $OC(B_p(3))$ if and only if $G{\cong}B_p(3)\;or\;C_p(3)$.

Keywords

References

  1. S. H. Alavi and A. Daneshkhah, A new characterization of alternating and symmetric groups, J. Appl. Math. Comput. 17 (2005), no. 1-2, 245-258 https://doi.org/10.1007/BF02936052
  2. G. Y. Chen, A new characterization of sporadic simple groups, Algebra Colloq. 3 (1996), no. 1, 49-58
  3. G. Y. Chen, A new characterization of $PSL_{2}(q)$, Southeast Asian Bull. Math. 22 (1998), no. 3, 257-263
  4. G. Y. Chen, On Thompson's conjecture, J. Algebra 185 (1996), no. 1, 184-193 https://doi.org/10.1006/jabr.1996.0320
  5. G. Y. Chen, On Frobenius and 2-Frobenius groups, J. Southwest China Normal Univ. 20 (1995), no. 5, 485-487
  6. G. Y. Chen and H. Shi, $^2D_n(3)9{\leq}n=2^m$ + 1 not a prime) can be characterized by its order components, J. Appl. Math. Comput. 19 (2005), no. 1-2, 353-362 https://doi.org/10.1007/BF02935810
  7. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985
  8. P. Crescenzo, A Diophantine equation which arises in the theory of finite groups, Advances in Math. 17 (1975), no. 1, 25-29 https://doi.org/10.1016/0001-8708(75)90083-3
  9. A. Iranmanesh, S. H. Alavi, and B. Khosravi, A characterization of PSL(3, q) where q is an odd prime power, J. Pure Appl. Algebra 170 (2002), no. 2-3, 243-254 https://doi.org/10.1016/S0022-4049(01)00113-X
  10. A. Iranmanesh, S. H. Alavi, and B. Khosravi, A characterization of PSL(3,q) for $q =2^{m}$, Acta Math. Sin. (Engl. Ser.) 18 (2002), no. 3, 463-472 https://doi.org/10.1007/s101140200164
  11. A. Iranmanesh, B. Khosravi, and S. H. Alavi, A characterization of $PSU_{3}$(q) for q > 5, Southeast Asian Bull. Math. 26 (2002), no. 1, 33-44 https://doi.org/10.1007/s100120200024
  12. A. Khosravi and B. Khosravi, A new characterization of PSL(p,q), Comm. Algebra 32 (2004), no. 6, 2325-2339 https://doi.org/10.1081/AGB-120037223
  13. A. Khosravi and B. Khosravi, A new characterization of some alternating and symmetric groups, Int. J. Math. Math. Sci. (2003), no. 45, 2863-2872
  14. A. Khosravi and B. Khosravi, r-recognizability of $B_{n}(q)$ and $C_{n}(q)$ where $n =2^{m}\;{\geq}\;4$, J. Pure Appl. Algebra 199 (2005), no. 1-3, 149-165 https://doi.org/10.1016/j.jpaa.2004.11.010
  15. Behrooz Khosravi, Behnam Khosravi, and Bahman Khosravi, The number of isomorphism classes of finite groups with the set of order components of $C_{4}(q)$, Appl. Algebra Engrg. Comm. Comput. 15 (2005), no. 5, 349-359 https://doi.org/10.1007/s00200-004-0166-4
  16. A. S. Kondratev, On prime graph components of finite simple groups, Mat. Sb. 180 (1989), no. 6, 787-797
  17. H. Shi and G. Y. Chen, $^{2}D_{p+1}$(2)(5${\leq}p{\neq}2^{m}-1$) can be characterized by its order components, Kumamoto J. Math. 18 (2005), 1-8
  18. J. S. Williams, Prime graph components of finite groups, J. Algebra 69 (1981), no. 2, 487-513 https://doi.org/10.1016/0021-8693(81)90218-0
  19. K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math. Phys. 3 (1892), no. 1, 265-284 https://doi.org/10.1007/BF01692444

Cited by

  1. CHARACTERIZATION OF THE GROUPS Dp+1(2) AND Dp+1(3) USING ORDER COMPONENTS vol.47, pp.2, 2010, https://doi.org/10.4134/JKMS.2010.47.2.311
  2. Characterizability of the group 2 D p(3) by its order components, where p ≥ 5 is a prime number not of the form 2 m + 1 vol.24, pp.7, 2008, https://doi.org/10.1007/s10114-007-6143-7
  3. A characterization of the group 2 D n (2), where n=2 m +1≥5 vol.31, pp.1-2, 2009, https://doi.org/10.1007/s12190-008-0223-4