Sylow like theorems for V (ℤG)

Kimmerle, Wolfgang

doi:10.22108/ijgt.2015.5452

تعداد نشریات	44
تعداد شماره‌ها	1,877
تعداد مقالات	15,278
تعداد مشاهده مقاله	43,731,781
تعداد دریافت فایل اصل مقاله	17,564,048

	Sylow like theorems for V (ℤG)
International Journal of Group Theory
مقاله 4، دوره 4، شماره 4، اسفند 2015، صفحه 49-59 اصل مقاله (212.6 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2015.5452
نویسنده
Wolfgang Kimmerle^*
University of Stuttgart
چکیده
‎The main part of this article is a survey on torsion subgroups of the unit group of‎ ‎an integral group ring‎. ‎It contains the major parts of my talk given at the‎ ‎conference‎ ‎"Groups,‎ ‎Group Rings and Related Topics‎" ‎at UAEU in Al Ain October 2013‎. ‎In the second part special emphasis is layed on $p$‎ - ‎subgroups and on the‎ ‎open question whether there is a Sylow like theorem in the‎ ‎normalized unit group of an integral group ring‎. ‎For specific classes of finite groups we prove that $p$‎ - ‎subgroups‎ ‎of the normalized unit group of its integral group rings $V(\mathbb{Z}G)$ ‎are isomorphic to subgroups of $G‎ .‎$ In particular for $p = 2$ this is shown ‎when $G$ has abelian Sylow $2$‎ - ‎subgroups‎. ‎This extends results known‎ ‎for soluble groups to classes of groups which are not contained in the‎ ‎class of soluble groups‎.
کلیدواژه‌ها
Group rings؛ integral group ring؛ Subgroup isomorphism problem

مراجع
[1] H. Ab old and W. Plesken, Ein Sylowsatz fur endliche $p$ -Untergrupp en von $GL (n,Z)$, Math. Ann., 232 (1978) 183-186. [2] A. Bachle and W. Kimmerle, On torsion subgroups in integral group rings of nite groups, J. Algebra, 326 (2011) 34-46. [3] A. Bachle and L. Margolis, Rational conjugacy of torsion units in integral group rings of non-solvable groups, math.RT, 2014, http://arXiv:1305.7419v2 . [4] S. D. Berman, On certain prop erties of integral group rings (Russian), Doklady Akad. Nauk SSSR (N.S.), 91 (1953) 7-9. [5] T. Breuer, GAP-package CTblLib The GAP Character Table Library, Version 1.2.1, 2012, http://www.math.rwth-aachen.de/ Thomas.Breuer/ctbllib . [6] V. Bovdi and M. Hertweck, Zassenhaus conjecture for central extensions of $S_5$ , J. Group Theory, 11 no.1 (2008) 63-74. [7] V. Bovdi, C. Hofert and W. Kimmerle, On the rst Zassenhaus conjecture for integral group rings, Publ. Math. Debrecen, 65 no. 3-4 (2004) 291-303. [8] V. Bovdi, E. Jesp ers and A. Konovalov, Torsion units in integral group rings of Janko simple groups, Math. Comp., 80 no. 273 (2011) 593-615. [9] M. Caicedo, A. del Rio and L. Margolis, Zassenhaus conjecture for cyclic-by-abelian groups, J. London Math. Soc. (2) , 88 (2013) 65-78. [10] J. A. Cohn and D. Livingstone, On the structure of group algebras I, Canad. J. Math., 17 (1965) 583-593. [11] M. A. Dokuchaev and S. O. Juriaans, Finite subgroups in integral group rings, Canad. J. Math., 48 no. 6 (1996) 1170-1179. [12] M. A. Dokuchaev, S. O. Juriaans and C. Polcino Milies, Integral group rings of Frobenius groups and the conjectures of H. J. Zassenhaus, Comm. Algebra, 25 no.7 (1997) 2311-2325. [13] The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.5.6, 2012, http://www.gap-system.org. [14] M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math.(2), 154 no. 1 (2001) 115-138. [15] M. Hertweck, On the torsion units of some integral group rings, Algebra Col loq., 13 no. 2 (2006) 329-348. [16] M. Hertweck, Partial augmentations and Brauer character values of torsion units in group rings, Preprint, 2007, http://arXiv:math.RA/0612429 . [17] M. Hertweck, Unit groups of integral nite group rings with nononcyclic abelian nite $p$-subgroups, Comm. Algebra, 36 no. 9 (2008) 3224-3229. [18] M. Hertweck, The orders of torsion units in integral group rings of nite solvable groups, Comm. Algebra, 36 no. 10 (2008) 3585-3588. [19] M. Hertweck, C. Hofert and W. Kimmerle, Finite groups of units and their comp osition factors in the integral group rings of the groups $PSL (2,q)$, J. Group Theory, 12 no. 6 (2009) 873-882. [20] M. Hertweck and W. Kimmerle, On principal blocks of p -constrained groups, Proc. London Math. Soc. (3), 84 no. 1 (2002) 179-193. [21] G. Higman, Units in group rings, D. Phil. thesis, Oxford Univ., 1940. [22] E. Jesp ers, W. Kimmerle, G. Nebe and Z. Marciniak, Miniworkshop Arithmetik von Gruppenringen,Oberwolfach Rep orts, EMS, Zurich, 4 no. 55 (2007) 3149-3179 . [23] S. O. Juriaans and C. Polcino Milies, Units of integral group rings of Frob enius groups, J. Group Theory, 3 no.3 (2000) 277-284. [24] W. Kimmerle, On the prime graph of the unit group of integral group rings of nite groups, Contemp. Math., 420, Amer. Math. Soc., Providence, RI, 2006 215-228. [25] W. Kimmerle, Torsion units in integral group rings of nite insoluble groups, 3169-3170. [26] W. Kimmerle, Unit Groups of Integral Group Rings: Old and New, Jahresber. Dtsch. Math.-Ver., 115 no. 2 (2013) 101-112. [27] W. Kimmerle and A. Konovalov, On the prime graph of the unit group of integral group rings of nite groups II, Stuttgarter Mathematische Berichte, 2012-018, http://www.mathematik.uni-stuttgart.de/fachb ereich/math-b erichte/listen.jsp. [28] W. Kimmerle and A. Konovalov, Recent advances on torsion subgroups of integral group rings, GroupsStAndrews, 2013, London Math. Soc. Lecture Note Series, 422, Cambridge University Press, (2015) 331-347. [29] W. Kimmerle and K. W. Roggenkamp, A Sylowlike theorem for integral group rings of nite solvable groups, Arch. Math. (Basel), 60 no.1 (1993) 1-6. [30] L. Klingler, Construction of a counterexample to a conjecture of Zassenhaus, Comm. Algebra, 19 no.8 (1991) 2303-2330. [31] I. S. Luthar and I. B. S. Passi, Zassenhaus conjecture for $A_5$, Indian Acad. Sci. Math. Sci., 99 no. 1(1989) 1-5. [32] Z. S. Marciniak, J. Ritter, S. K. Sehgal and A. Weiss, Torsion Units in Integral Group Rings of Some Metab elian Groups, II, J. Number Theory, 25 (1987) 340-352. [33] K. W. Roggenkamp, Observations to a conjecture of H. Zassenhaus, Groups St. Andrews 1989, London Math. Soc. Lecture Note Series, 160 no. 2 (1991) 427-444. [34] K. W. Roggenkamp, The isomorphism problem for integral group rings of nite groups, Proceedings of the Interna-tional Congress of Mathematicians, I, II, (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991 369-380. [35] K. W. Roggenkamp and L. L. Scott, Isomorphisms of $p$-adic group rings, Ann. of Math., 126 (1987) 593-647. [36] K. W. Roggenkamp and L. L. Scott, A strong answer to the isomorphism problem for nite $p$-solvable groups with a normal subgroup containing its centralizer, Manuscript 1987. [37] K. W. Roggenkamp and L. L. Scott, On a conjecture of Zassenhaus for integral group rings of nitegroups, Manu-script 1988. [38] M. A. Salim, The prime graph conjecture for integral group rings of some alternating groups, Int. J. Group Theory, 2 no.1 (2013) 175-185. [39] S. K. Sehgal, Units in integral group rings , Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientic & Technical, Essex, 1993. [40] L. L. Scott, Recent progress on the isomorphism problem, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Pro c. Symp os. Pure Math., 47 , Part 1, Amer. Math. Soc., Providence, RI, 1987 259-273. [41] L. L. Scott, On a conjecture of Zassenhaus, and beyond, Proceedings of the International Conference on Algebra, Part 1 (Novosibirsk, 1989), Contemp. Math., 131, Part 1, Amer. Math. Soc., Providence, RI, 1992 325{343. [42] J. H. Walter, The characterization of nite groups with Abelian Sylow $2$-subgroups, Ann. of Math.(2), 89 (1969) 405-514. [43] A. Weiss, Rigidity of $p$-adic $p$-torsion, Ann. of Math. (2), 127 no. 2 (1987) 317-332. [44] A. Weiss, Torsion units in integral group rings, J. Reine Angew. Math., 415 (1991) 175-187. [45] H. Zassenhaus, On the torsion units of nite group rings, Studies in mathematics (in honor of A. Almeida Costa) (Portuguese), Instituto de Alta Cultura, Lisb on, 1974 119-126.
آمار تعداد مشاهده مقاله: 4,050 تعداد دریافت فایل اصل مقاله: 3,717

سامانه مدیریت نشریات علمی. طراحی و پیاده سازی از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

Sylow like theorems for V (ℤG)