![]() Hatcher, Algebraic topology, Cambridge University Press, Cambridge, (2002). Hamm - Lê Dũng Tráng, Rectified homotopical depth and Grothendieck conjectures, The Grothendieck Festschrift, Vol. Hamm - Lê Dũng Tráng, Local generalizations of Lefschetz-Zariski theorems, J. ![]() Hamm - Lê Dũng Tráng, Lefschetz theorems on quasi-projective varieties, Bull. Hamm - Lê Dũng Tráng, Un théorème de Zariski du type de Lefschetz, Ann. Hamm, Connectedness of the Milnor fibre and Stein factorization of compactifiable holomorphic functions, Preprint (2019). Hamm, Zum Homotopietyp q-vollständiger Räume, J. ![]() Hamm, Lefschetz theorems for singular varieties, in Proc. Hamm, Zum Homotopietyp Steinscher Räume, J. Hamm, Lokale topologische Eigenschaften komplexer Räume, Math. North-Holland Publishing Co., Amsterdam Masson & Cie, Éditeur, Paris, (1968). Advanced Studies in Pure Mathematics, Vol. augmenté d’un exposé par Michèle Raynaud, Séminaire de Géométrie Algébrique du Bois-Marie, 1962. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2). MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 14, Springer-Verlag, Berlin, (1988) xiv + 272 pp.Ī. Gonzalez-Sprinberg, Une formule pour les singularités isolées de surfaces, C. ![]() In: Algebraic geometry (Chicago, Ill., 1980), Lecture Notes in Math. Lazarsfeld, Connectivity and its applications in algebraic geometry. Kaup, Theorems of Lefschetz type in intersection homology. Eyral, Profondeur homotopique et conjecture de Grothendieck, Ann. 1–10, Berlin Heidelberg New York, Springer (1981).Ĭ. Deligne, Le groupe fondamental du complémentaire d’une courbe plane n’ayant que des points doubles ordinaires est abélien (d’après Fulton) in Sém. Cheniot, Topologie du complémentaire d’un ensemble algébrique projectif, Enseign. Enriques, Sur les intégrales simples de première espèce d’une surface ou d’une variété algébrique à plusieurs dimensions, Ann. Verona, Local homological properties of analytic sets, Manuscripta Math. Bott, On a theorem of Lefschetz, Michigan Math. Larsen, On the homotopy groups of complex projective algebraic manifolds, Math. Frankel, The Lefschetz theorem on hyperplane sections, Ann. We also see that the index p is also a divisor of the order of the group.A. This shows that n, the order of H, is a divisor of m, the order of the finite group G. So, the total number of elements of all cosets is np which is equal to the total number of elements of G. Since G is a finite group, the number of discrete left cosets will also be finite, say p. Suppose, ahi=ahj⇒hi=hj be the cancellation law of G. Thus, the subgroups of G will be, then ah1,ah2,…,ahn are the n distinct members of aH. Now, m will have only two divisors 1 and m (prime numbers property). Proof: Let us suppose, the prime order of group G is m. So, we can write, m = np, where n is a positive integer.Ĭorollary 2: If the order of finite group G is a prime order, then it does not have proper subgroups. Since the subgroup has order p, thus p the order of a is the divisor of group G. Proof: Let the order of a be p, which is the least positive integer, so,Ī, a 2, a 3, …., a p-1,a p = e, the elements of group G are all different and they form a subgroup. Let us now prove some corollaries relating to Lagrange's theorem.Ĭorollary 1: If G is a group of finite order m, then the order of any a∈G divides the order of G and in particular a m = e.
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