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Keywords:
almost simple group; prime graph; degree of a vertex; degree pattern
almost simple group; prime graph; degree of a vertex; degree pattern
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 $G\cong M$.
References:
[1] Chen, G. Y.: On structure of Frobenius and $2$-Frobenius group. J. Southwest China Normal Univ. 20 (5) (1995), 485–487, (in Chinese).
[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).
[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. MR 0827219 | Zbl 0568.20001
[4] Gorenstein, D.: Finite Groups. Harper and Row, New York, 1980. MR 0569209 | Zbl 0463.20012
[5] Higman, G.: Finite groups in which every element has prime power order. J. London Math. Soc. 32 (1957), 335–342. DOI 10.1112/jlms/s1-32.3.335 | MR 0089205 | Zbl 0079.03204
[6] Iiyori, N.: Sharp charaters and prime graphs of finite groups. J. Algebra 163 (1994), 1–8. DOI 10.1006/jabr.1994.1001 | MR 1257302
[7] Kondratev, A. S.: On prime graph components of finite simple groups. Mat. Sb. 180 (6) (1989), 787–797. MR 1015040
[8] Mazurov, V. D.: The set of orders of elements in a finite group. Algebra and Logic 33 (1) (1994), 49–55. DOI 10.1007/BF00739417 | MR 1287011 | Zbl 0823.20024
[9] Mazurov, V. D.: Characterizations of finite groups by sets of orders of their elements. Algebra and Logic 36 (1) (1997), 23–32. DOI 10.1007/BF02671951 | MR 1454690 | Zbl 0880.20007
[10] Moghaddamfar, A. R., Zokayi, A. R.: Recognizing finite groups through order and degree pattern. to appear in Algebra Colloquium.
[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. MR 2144997 | Zbl 1072.20015
[12] Passman, D.: Permutation Groups. Benjamin Inc., New York, 1968. MR 0237627 | Zbl 0179.04405
[13] Shi, W. J.: A new characterization of some simple groups of Lie type. Contemp. Math. 82 (1989), 171–180. DOI 10.1090/conm/082/982286 | MR 0982286 | Zbl 0668.20019
[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
[15] Shi, W. J.: Pure quantitive characterization of finite simple groups (I). Progr. Natur. Sci. (English Ed.) 4 (3) (1994), 316–326. MR 1402664
[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. MR 1092230 | Zbl 0718.20009
[17] Williams, J. S.: Prime graph components of finite groups. J. Algebra 69 (2) (1981), 487–513. DOI 10.1016/0021-8693(81)90218-0 | MR 0617092 | Zbl 0471.20013
[18] Yamaki, H.: A conjecture of Frobenius and the sporadic simple groups I. Comm. Algebra 11 (1983), 2513–2518. DOI 10.1080/00927878308822977 | MR 0733339
[19] Yamaki, H.: A conjecture of Frobenius and the simple groups of Lie type I. Arch. Math. 42 (1984), 344–347. DOI 10.1007/BF01246125 | MR 0753355 | Zbl 0528.20024
[20] Yamaki, H.: A conjecture of Frobenius and the simple groups of Lie type II. J. Algebra 96 (1985), 391–396. DOI 10.1016/0021-8693(85)90017-1 | MR 0810536 | Zbl 0572.20013
[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. MR 0829631 | Zbl 0585.20008
[22] Zhang, L. C., Shi, W. J.: $OD$-characterization of almost simple groups related to $U_{4}(3)$. to appear. MR 2425165
[23] Zhang, L. C., Shi, W. J.: $OD$-characterization of simple $K_{4}$-groups. to appear in Algebra Colloquium (in press).
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