Abstract
In representation theory of finite groups, one of the most important and interesting problems is that, for a p-block A of a finite group G where p is a prime, the numbers k(A) and ℓ(A) of irreducible ordinary and Brauer characters, respectively, of G in A are p-locally determined. We calculate k(A) and ℓ(A) for the cases where A is a full defect p-block of G, namely, a defect group P of A is a Sylow p-subgroup of G and P is a nonabelian metacyclic p-group M n+1(p) of order p n+1 and exponent p n for \({n \geqslant 2}\), and where A is not necessarily a full defect p-block but its defect group P = M n+1(p) is normal in G. The proof is independent of the classification of finite simple groups.
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References
Alperin J.L.: Local Representation Theory. Cambridge University Press, Cambridge (1986)
J. L. Alperin, Weights for finite groups, in: The Arcata Conference on Representations of Finite Groups, P. Fong (ed.), Proc. Symposia Pure Math., 47/1, 369–379, Amer. Math. Soc., Providence 1987.
Alperin J.L., Linckelmann M., Rouquier R.: Source algebras and source modules. J. Algebra 239, 262–271 (2001)
M. Broué, Isométries parfaites, types de blocs, catégories dérivées, Astérisque 181–182 (1990), 61–92.
Conway J.H. et al.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)
Dade E.C.: Blocks with cyclic defect groups. Ann. of Math. 84, 20–48 (1966)
Dornhoff L.: Group Representation Theory (Part B). Marcel Dekker, New York (1972)
Gorenstein D.: Finite Groups. Harper and Row, New York (1968)
Hendren S.: Extra special defect groups of order p 3 and exponent p 2. J. Algebra 291, 457–491 (2005)
M. Holloway, Broué’s conjecture for the Hall-Janko group and its double cover, Proc. London Math. Soc. (3) 86 (2003), 109–130.
Huppert B.: Endliche Gruppen I. Springer-Verlag, Berlin (1967)
Huppert B., Blackburn N.: Finite Groups II. Springer-Verlag, Berlin (1982)
Jansen C. et al.: An Atlas of Brauer Characters. Clarendon Press, Oxford (1995)
Knörr R.: On the vertices of irreducible modules. Ann. of Math. 110, 487–499 (1979)
Koshitani S., Külshammer B.: A splitting theorem for blocks. Osaka J. Math. 33, 343–346 (1966)
Külshammer B.: Crossed products and blocks with normal defect groups. Commun. Algebra 13, 147–168 (1985)
Linckelmann M.: Le centre d’un bloc à groupes de défaut cycliques. C. R. Acad. Sci. Paris. 306, 727–730 (1988)
Nagao H., Tsushima Y.: Representations of Finite Groups. Academic Press, New York (1988)
Puig L.: Pointed groups and construction of modules. J. Algebra 116, 7–129 (1988)
Puig L.: The hyperfocal subalgebra of a block. Invent. Math. 141, 365–397 (2000)
Puig L.: Blocks of Finite Groups. Springer-Verlag, Berlin (2002)
Rickard J.: Equivalences of derived categories for symmetric algebras. J. Algebra 257, 460–481 (2002)
R. Rouquier, From stable equivalences to Rickard equivalences for blocks with cyclic defect, in: Groups ’93 Galway/St Andrews, C. M. Campbell et al. (eds.), London Mathematical Society Lecture Note Series 212, 512–523 Cambridge University Press, Cambridge 1995.
Sasaki H.: The mod p cohomology algebras of finite groups with metacyclic Sylow p-subgroups. J. Algebra 192, 713–733 (1997)
Suzuki M.: Group Theory II. Springer, Berlin (1986)
Thévenaz J.: G-Algebras and Modular Representation Theory. Clarendon Press, Oxford (1995)
Wong W.J.: On finite groups whose 2-Sylow subgroups have cyclic subgroups of index 2. J. Australian Math. Soc. 4, 90–112 (1964)
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Holloway, M., Koshitani, S. & Kunugi, N. Blocks with nonabelian defect groups which have cyclic subgroups of index p . Arch. Math. 94, 101–116 (2010). https://doi.org/10.1007/s00013-009-0075-7
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DOI: https://doi.org/10.1007/s00013-009-0075-7