Abstract
In this chapter we are able to determine the invariants of block whose defect groups are direct and central products of metacyclic 2-groups. More precisely, we consider products of cyclic groups and 2-groups of maximal nilpotency class. These are the dihedral, semidihedral and quaternion groups. As an application we verify several open conjectures for these special cases.
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References
Brauer, R.: On 2-blocks with dihedral defect groups. In: Symposia Mathematica, vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), pp. 367–393. Academic, London (1974)
Broué, M.: Equivalences of blocks of group algebras. In: Finite-Dimensional Algebras and Related Topics (Ottawa, ON, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 424, pp. 1–26. Kluwer Academic Publishers, Dordrecht (1994)
Cabanes, M., Picaronny, C.: Types of blocks with dihedral or quaternion defect groups. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 39(1), 141–161 (1992). Revised version: http://www.math.jussieu.fr/~cabanes/type99.pdf
Gorenstein, D.: Finite groups. Harper & Row Publishers, New York (1968)
Harada, K.: Groups with a certain type of Sylow 2-subgroups. J. Math. Soc. Jpn. 19, 203–307 (1967)
Linckelmann, M.: Introduction to fusion systems. In: Group Representation Theory, pp. 79–113. EPFL Press, Lausanne (2007). Revised version: http://web.mat.bham.ac.uk/C.W.Parker/Fusion/fusion-intro.pdf
Olsson, J.B.: On 2-blocks with quaternion and quasidihedral defect groups. J. Algebra 36(2), 212–241 (1975)
Olsson, J.B.: Lower defect groups. Commun. Algebra 8(3), 261–288 (1980)
Park, S.: The gluing problem for some block fusion systems. J. Algebra 323(6), 1690–1697 (2010)
Puig, L., Usami, Y.: Perfect isometries for blocks with abelian defect groups and cyclic inertial quotients of order 4. J. Algebra 172(1), 205–213 (1995)
Sambale, B.: Cartan matrices and Brauer’s k(B)-conjecture II. J. Algebra 337, 345–362 (2011)
Sambale, B.: Blocks with defect group \(D_{2^{n}} \times C_{2^{m}}\). J. Pure Appl. Algebra 216, 119–125 (2012)
Sambale, B.: Blocks with central product defect group \(D_{2^{n}} {\ast} C_{2^{m}}\). Proc. Am. Math. Soc. 141(12), 4057–4069 (2013)
Sambale, B.: Blocks with defect group \(Q_{2^{n}} \times C_{2^{m}}\) and \(SD_{2^{n}} \times C_{2^{m}}\). Algebr. Represent. Theory 16(6), 1717–1732 (2013)
Usami, Y.: On p-blocks with abelian defect groups and inertial index 2 or 3. I. J. Algebra 119(1), 123–146 (1988)
Wang, B.: Modular representations of direct products. MM Res. Preprints 22, 256–263 (2003)
Webb, P.: An introduction to the representations and cohomology of categories. In: Group Representation Theory, pp. 149–173. EPFL Press, Lausanne (2007)
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Sambale, B. (2014). Products of Metacyclic Groups. In: Blocks of Finite Groups and Their Invariants. Lecture Notes in Mathematics, vol 2127. Springer, Cham. https://doi.org/10.1007/978-3-319-12006-5_9
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DOI: https://doi.org/10.1007/978-3-319-12006-5_9
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