On 2-blocks with semidihedral defect groups

Erdmann, Karin

doi:10.1090/S0002-9947-1979-0546918-4

On $2$-blocks with semidihedral defect groups
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Trans. Amer. Math. Soc. 256 (1979), 267-287 Request permission

Abstract:

Let G be one of the following groups: ${L_3}(q)$ and $GL(2, q)$ with $q \equiv 3 (\bmod 4)$, ${U_3}(q)$ and $GU(2, q)$ with $q \equiv 1 (\bmod 4)$. This paper is concerned with 2-blocks B of G having semidihedral defect groups. In particular, vertices, sources and Green correspondents of the simple modules in B are determined and used to obtain the submodule structure of the indecomposable projective modules.

References

J. L. Alperin, Richard Brauer, and Daniel Gorenstein, Finite groups with quasi-dihedral and wreathed Sylow $2$-subgroups, Trans. Amer. Math. Soc. 151 (1970), 1–261. MR 284499, DOI 10.1090/S0002-9947-1970-0284499-5
Larry Dornhoff, Group representation theory. Part A: Ordinary representation theory, Pure and Applied Mathematics, vol. 7, Marcel Dekker, Inc., New York, 1971. MR 0347959
Karin Erdmann, Principal blocks of groups with dihedral Sylow $2$-subgroups, Comm. Algebra 5 (1977), no. 7, 665–694. MR 447391, DOI 10.1080/00927877708822186
Karin Erdmann, Blocks and simple modules with cyclic vertices, Bull. London Math. Soc. 9 (1977), no. 2, 216–218. MR 444761, DOI 10.1112/blms/9.2.216
William A. Simpson and J. Sutherland Frame, The character tables for $\textrm {SL}(3,\,q)$, $\textrm {SU}(3,\,q^{2})$, $\textrm {PSL}(3,\,q)$, $\textrm {PSU}(3,\,q^{2})$, Canadian J. Math. 25 (1973), 486–494. MR 335618, DOI 10.4153/CJM-1973-049-7
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903
J. A. Green, Relative module categories for finite groups, J. Pure Appl. Algebra 2 (1972), 371–393. MR 384913, DOI 10.1016/0022-4049(72)90012-6
J. A. Green, Vorlesungen über Modulare Darstellungstheorie endlicher Gruppen, Vorlesungen aus dem Mathematischen Institut Giessen, Heft 2, Universität Giessen, Mathematisches Institut, Giessen, 1974 (German). An der Universität Giessen im Sommersemester 1974; With an English acknowledgment; Manuskript: Wolfgang Hamernik. MR 0360788
B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703, DOI 10.1007/978-3-642-64981-3

On the vertices of simple modules

Wolfgang Müller, Gruppenalgebren über nichtzyklischen $p$-Gruppen. II, J. Reine Angew. Math. 267 (1974), 1–19 (German). MR 344325, DOI 10.1515/crll.1974.267.1
Jørn Børling Olsson, On $2$-blocks with quaternion and quasidihedral defect groups, J. Algebra 36 (1975), no. 2, 212–241. MR 376841, DOI 10.1016/0021-8693(75)90099-X
L. Scott, The modular theory of permutation representations, Representation theory of finite groups and related topics (Proc. Sympos. Pure Math., Vol. XXI, Univ. Wisconsin, Madison, Wis., 1970) Amer. Math. Soc., Providence, R.I., 1971, pp. 137–144. MR 0320133