APM  Vol.5 No.4 , March 2015
On k(D)-Blocks
Author(s) Ahmad M. Alghamdi*
ABSTRACT
The objective of this research paper is to study numerical relationships between a block of a finite group and a defect group of such block. We define a new notion which is called a strongly k(D)- block and give a necessary and sufficient condition of a block with a cyclic defect group to be a k(D) -block in term of its inertial index. We believe that the notion and the results in this work will contribute to the developments of the theory of blocks of finite groups.

Cite this paper
Alghamdi, A. (2015) On k(D)-Blocks. Advances in Pure Mathematics, 5, 150-154. doi: 10.4236/apm.2015.54018.
References
[1]   Alghamdi, A. (2004) The Ordinary Weight Conjecture and Dade’s Projective Conjecture for p-Blocks with an Extra Special Defect Group. Ph.D. Dissertation, University of Birmingham, Birmingham.

[2]   Brauer, R. and Feit, W. (1959) On the Number of Irreducible Characters of Finite Groups in a Given Block. Proceedings of the National Academy of Sciences of the United States of America, 45, 361-365.
http://dx.doi.org/10.1073/pnas.45.3.361

[3]   Robinson, G.R. (1997) Some Open Conjectures on Representation Theory. In: Columbus, O.H., Ed., Representation Theory of Finite Groups, Ohio State Univ. Math. Res. Inst. Publ., de Gruyter, Berlin, 6, 127-131.

[4]   Robinson, G.R. (1992) On Brauer’s k(B) Problem. Journal of Algebra, 147, 450-455.
http://dx.doi.org/10.1016/0021-8693(92)90215-8

[5]   Knorr, R. (1984) On the Number of Characters in a p-Block of a p-Solvable Group. Illinois Journal of Mathematics, 28, 181-210.

[6]   Kulshammer, B. and Robinson, G. (1996) Alperin-Makey Implies Brauer’s Problem 21. Journal of Algebra, 180, 208210.
http://dx.doi.org/10.1006/jabr.1996.0062

[7]   Kulshammer, B. (1996) Modular Representations of Finite Groups: Conjectures and Examples, Jena.

[8]   Dade, E. (1996) Counting Characters in Blocks with Cyclic Defect Groups, I. Journal of Algebra, 186, 934-969.
http://dx.doi.org/10.1006/jabr.1996.0401

[9]   Dornhoff, L. (1972) Group Representation Theory, Part B: Modular Representation Theory. Marcel Dekker Inc., New York.

[10]   Kulshammer, B. and Sambale, B. (2013) The 2-Blocks of Defect 4. Journal of Representation Theory, 17, 226-236.

[11]   Kessar, R. and Malle, G. (1992) Quasi-Isoloated Blocks and Height Zero Conjecture. Journal of Algebra, 147, 450455.

[12]   Nagao, H. and Tsushima, Y. (1989) Representation of Finite Groups. Academic Press Inc., Boston, Translated from Japanese.

[13]   Dade, E. (1966) Blocks with Cyclic Defect Groups. Annals of Mathematics, Second Series, 84, 20-48.
http://dx.doi.org/10.2307/1970529

[14]   Navarro, G. (1998) Characters and Blocks of Finite Groups, Volume 250 of London Mathematical Society Lecture Notes Series. Cambridge University Press, Cambridge.

[15]   Brauer, R. (1956) Zur Darstellungstheori der Gruppen endlicher Ordnung I. Mathematische Zeitschrift, 63, 406-444.
http://dx.doi.org/10.1007/BF01187950

[16]   Brauer, R. (1959) Zur Darstellungstheori der Gruppen endlicher Ordnung II. Mathematische Zeitschrift, 72, 25-46.
http://dx.doi.org/10.1007/BF01162934

[17]   Brauer, R. (1964) Some Applications of the Theory of Blocks of Characters of Finite Groups I. Journal of Algebra, 1, 152-167.
http://dx.doi.org/10.1016/0021-8693(64)90031-6

[18]   Brauer, R. (1964) Some Applications of the Theory of Blocks of Characters of Finite Groups II. Journal of Algebra, 1, 307-334.
http://dx.doi.org/10.1016/0021-8693(64)90011-0

[19]   Brauer, R. (1966) Some Applications of the Theory of Blocks of Characters of Finite Groups III. Journal of Algebra, 3, 225-255.
http://dx.doi.org/10.1016/0021-8693(66)90013-5

[20]   Brauer, R. (1971) Some Applications of the Theory of Blocks of Characters of Finite Groups IV. Journal of Algebra, 17, 489-521.
http://dx.doi.org/10.1016/0021-8693(71)90006-8

[21]   Brauer, R. (1974) Some Applications of the Theory of Blocks of Characters of Finite Groups V. Journal of Algebra, 28, 433-460.
http://dx.doi.org/10.1016/0021-8693(74)90051-9

 
 
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