| Title:
|
OD-characterization of almost simple groups related to $L_{2}(49)$ (English) |
| Author:
|
Zhang, Liangcai |
| Author:
|
Shi, Wujie |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
44 |
| Issue:
|
3 |
| Year:
|
2008 |
| Pages:
|
191-199 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the present paper, we classify groups with the same order and degree pattern as an almost simple group related to the projective special linear simple group $L_{2}(49)$. As a consequence of this result we can give a positive answer to a conjecture of W. J. Shi and J. X. Bi, for all almost simple groups related to $L_{2}(49)$ except $L_{2}(49)\cdot 2^{2}$. Also, we prove that if $M$ is an almost simple group related to $L_{2}(49)$ except $L_{2}(49)\cdot 2^{2}$ and $G$ is a finite group such that $|G|=|M|$ and $\Gamma (G)=\Gamma (M)$, then $G\cong M$. (English) |
| Keyword:
|
almost simple group |
| Keyword:
|
prime graph |
| Keyword:
|
degree of a vertex |
| Keyword:
|
degree pattern |
| MSC:
|
20D05 |
| MSC:
|
20D06 |
| MSC:
|
20D60 |
| idZBL:
|
Zbl 1204.20006 |
| idMR:
|
MR2462974 |
| . |
| Date available:
|
2009-01-29T09:14:51Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119758 |
| . |
| Reference:
|
[1] Chen, G. Y.: On structure of Frobenius and $2$-Frobenius group.J. Southwest China Normal Univ. 20 (5) (1995), 485–487, (in Chinese). |
| Reference:
|
[2] Chen, Z. M., Shi, W. J.: On $C_{p,p}$-simple groups.J. Southwest China Normal Univ. 18 (3) (1993), 249–256, (in Chinese). |
| Reference:
|
[3] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A.: Atlas of Finite Groups.Clarendon Press (Oxford), London – New York, 1985. Zbl 0568.20001, MR 0827219 |
| Reference:
|
[4] Gorenstein, D.: Finite Groups.Harper and Row, New York, 1980. Zbl 0463.20012, MR 0569209 |
| Reference:
|
[5] Higman, G.: Finite groups in which every element has prime power order.J. London Math. Soc. 32 (1957), 335–342. Zbl 0079.03204, MR 0089205, 10.1112/jlms/s1-32.3.335 |
| Reference:
|
[6] Iiyori, N.: Sharp charaters and prime graphs of finite groups.J. Algebra 163 (1994), 1–8. MR 1257302, 10.1006/jabr.1994.1001 |
| Reference:
|
[7] Kondratev, A. S.: On prime graph components of finite simple groups.Mat. Sb. 180 (6) (1989), 787–797. MR 1015040 |
| Reference:
|
[8] Mazurov, V. D.: The set of orders of elements in a finite group.Algebra and Logic 33 (1) (1994), 49–55. Zbl 0823.20024, MR 1287011, 10.1007/BF00739417 |
| Reference:
|
[9] Mazurov, V. D.: Characterizations of finite groups by sets of orders of their elements.Algebra and Logic 36 (1) (1997), 23–32. Zbl 0880.20007, MR 1454690, 10.1007/BF02671951 |
| Reference:
|
[10] Moghaddamfar, A. R., Zokayi, A. R.: Recognizing finite groups through order and degree pattern.to appear in Algebra Colloquium. |
| Reference:
|
[11] Moghaddamfar, A. R., Zokayi, A. R., Darafsheh, M. R.: A characterization of finite simple groups by the degrees of vertices of their prime graphs.Algebra Colloq. 12 (3) (2005), 431–442. Zbl 1072.20015, MR 2144997 |
| Reference:
|
[12] Passman, D.: Permutation Groups.Benjamin Inc., New York, 1968. Zbl 0179.04405, MR 0237627 |
| Reference:
|
[13] Shi, W. J.: A new characterization of some simple groups of Lie type.Contemp. Math. 82 (1989), 171–180. Zbl 0668.20019, MR 0982286, 10.1090/conm/082/982286 |
| Reference:
|
[14] Shi, W. J.: A new characteriztion 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. MR 0981868 |
| Reference:
|
[15] Shi, W. J.: Pure quantitive characterization of finite simple groups (I).Progr. Natur. Sci. (English Ed.) 4 (3) (1994), 316–326. MR 1402664 |
| Reference:
|
[16] Shi, W. J., Bi, J. X.: A characteristic property for each finite projective special linear group.Lecture Notes in Math. 1456 (1990), 171–180Wi. Zbl 0718.20009, MR 1092230 |
| Reference:
|
[17] Williams, J. S.: Prime graph components of finite groups.J. Algebra 69 (2) (1981), 487–513. Zbl 0471.20013, MR 0617092, 10.1016/0021-8693(81)90218-0 |
| Reference:
|
[18] Yamaki, H.: A conjecture of Frobenius and the sporadic simple groups I.Comm. Algebra 11 (1983), 2513–2518. MR 0733339, 10.1080/00927878308822977 |
| Reference:
|
[19] Yamaki, H.: A conjecture of Frobenius and the simple groups of Lie type I.Arch. Math. 42 (1984), 344–347. Zbl 0528.20024, MR 0753355, 10.1007/BF01246125 |
| Reference:
|
[20] Yamaki, H.: A conjecture of Frobenius and the simple groups of Lie type II.J. Algebra 96 (1985), 391–396. Zbl 0572.20013, MR 0810536, 10.1016/0021-8693(85)90017-1 |
| Reference:
|
[21] Yamaki, H.: A conjecture of Frobenius and the sporadic simple groups II.Math. Comp. 46 (1986), 609–611, Supplement, Math. Comp., 46, 1986), S43-S46. Zbl 0585.20008, MR 0829631 |
| Reference:
|
[22] Zhang, L. C., Shi, W. J.: $OD$-characterization of almost simple groups related to $U_{4}(3)$.to appear. MR 2425165 |
| Reference:
|
[23] Zhang, L. C., Shi, W. J.: $OD$-characterization of simple $K_{4}$-groups.to appear in Algebra Colloquium (in press). |
| . |
