Recognizing the Non-Frattini Abelian Chief Factors of a Finite Group from Its Probabilistic Zeta Function

REFERENCES

• Boston , N. ( 1996 ). A probabilistic generalization of the Riemann zeta function . Anal. Num. Theory 1 : 155 – 162 .
   Google Scholar
• Dalla Volta , F. , Lucchini , A. , Morini , F. ( 2003 ). Some remarks on the probability of generating an almost simple group . Glasgow Math. J. 45 : 281 – 291 .
   Web of Science ®Google Scholar
• Damian , E. , Lucchini , A. ( 2003 ). The Dirichlet polynomial of a finite group and the subgroups of prime power index. In: Advances in Group Theory 2002. Rome: Aracne, pp. 209–221 .
   Google Scholar
• Damian , E. , Lucchini , A. ( 2007 ). The Probabilistic Zeta function of finite simple group . J. Algebra 313 : 957 – 971 .
   Web of Science ®Google Scholar
• Damian , E. , Lucchini , A. , Morini , F. ( 2004 ). Some properties of the Probabilistic Zeta function of finite simple groups . Pacific J. Math. 215 : 3 – 14 .
   Web of Science ®Google Scholar
• Detomi , E. , Lucchini , A. ( 2003 ). Crowns and factorization of the probabilistic zeta function of a finite group . J. Algebra 265 : 651 – 668 .
   Web of Science ®Google Scholar
• Detomi , E. , Lucchini , A. ( 2003 ). Recognizing soluble groups from their Probabilistic Zeta function . Bull. London Math. Soc. 35 : 659 – 664 .
   Web of Science ®Google Scholar
• Detomi , E. , Lucchini , A. (2007). Some generalization of the probabilistic zeta function. In: Ischia Group Theory 2006, pp. 56–72.
   Google Scholar
• Ecklund , E. F. , Eggleton , R. B. , Erdös , P. , Selfridge , J. L. ( 1978 ). On the prime factorization of binomial coefficients. J. Australian Math. Soc. (Series A) 26(3):257–269 .
   Google Scholar
• The GAP Group ( 2008 ). GAP—Groups, Algorithms, and Programming. Version 4.4.12 .
   Google Scholar
• Gaschütz , W. ( 1959 ). Die Eulersche Funktion endlicher auflösbarer Gruppen . Illinois J. Math. 3 : 469 – 476 .
   Google Scholar
• Gaschütz , W. ( 1962 ). Praefrattinigruppen . Arch. Math. 13 : 418 – 426 .
   Google Scholar
• Giudici , M. ( 2007 ). Maximal subgroups of almost simple groups with socle PSL(2, q). ArXiv Mathematics e-prints. arXiv:math/0703685v1 .
   Google Scholar
• Guralnick , R. ( 1983 ). Subgroups of prime power index in a simple group . J. Algebra 81 : 304 – 311 .
   Web of Science ®Google Scholar
• Hall , P. ( 1936 ). The Eulerian functions of a group . Quart. J. Math. 7 : 134 – 151 .
   Google Scholar
• Jordan , C. ( 1872–73 ). Sur la limite de transitivité des groupes non alternés . Bull. Soc. Math. France , pp. 40 – 71 .
   Google Scholar
• Jordan , C. ( 1957 ). Traité des Substitutions et des Équations Algébriques – Nouveau Tirage de l’édition de 1870 . Paris : Gauthier-Villars, A. Blanchard .
   Google Scholar
• Lang , S. ( 2002 ). Algebra . Vol. 211 , Graduate Texts in Mathematics, New York : Springer-Verlag .
   Google Scholar
• Mann , A. ( 1996 ). Positively finitely generated groups . Forum Math. 8 : 429 – 459 .
   Web of Science ®Google Scholar
• Patassini , M. ( 2011 ). On the irreducibility of the Dirichlet polynomial of a simple group of Lie type . Israel J. Math. 185 ( 1 ): 477 – 507 .
   Web of Science ®Google Scholar
• Patassini , M. ( 2011 ). Recognizing the characteristic of a group of Lie type from its Probabilistic Zeta function . J. Algebra 332 ( 1 ): 480 – 499 .
   Web of Science ®Google Scholar
• Jiménez Seral , P. ( 2008 ). Coefficient of the Probabilistic Zeta function of a monolithic group . Glasgow J. Math. 50 : 75 – 81 .
   Web of Science ®Google Scholar
• Communicated by P. Tiep.
   Google Scholar