Abstract
Group representation theory has grown out of group theory. Therefore one way of measuring progress is by looking at the applications of representation theory to group theory and theories where groups play a role. In this note I discuss two areas of applications, which I have been involved in. A third area, which is left out here, is that of extensions. As pointed out in [HoP 89] representation theory not only provides abelian normal subgroups for extensions with given factor group, but also nonabelian (e.g. pro-p) normal subgroups. A fourth area, which I also only mention in passing, is that of applications to computational group theory, more precisely to the investigation of finite presentations of groups, cf. [Ple 87], [HoP 89] Chapter 7, [HoP 90].
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ei. Bannai, Et. Bannai, On some finite subgroups of GL(n, Q). J. Fac. Sci. Univ. Tokyo, Sect. IA 20,(3) (1973), 319–340.
H. F. Blichfeldt, Finite Collineation Groups. Chicago: University of Chicago Press 1917.
R. Brauer, Über endliche lineare Gruppen von Primzahlgrad. Math. Annalen 169, 73–96 (1967).
H. Brown, R. Bülow, J. Neubüser, H. Wondratschek, H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley 1977.
R. Bülow, Über Dadegruppen in GL(5, Z). Dissertation RWTH Aachen 1973.
W. Burnside, The determination of all groups of rational linear substitutions of finite order which contain the symmetric group in the variables. Proc. London Math. Cos. (2) 10 (1912), 284–308.
P.J. Cameron, Finite Permutation Groups and Finite Simple Groups. Bull. London Math. Soc. 13 (1981), 1–22.
J.H. Conway, N.J.A. Sloane, Low-dimensional lattices. II. Subgroups of GL(n, Z). Proc. R. Soc. London A 419, 29–68 (1988).
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups. Oxford University Press 1985.
C.W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras. Interscience, New York, 1962.
E.C. Dade, Integral Systems of Imprimitivity. Math. Annalen 154, 383–386 (1964).
E.C. Dade, The maximal finite groups of 4 × 4 matrices. Ill. J. Math. 9 (1965), 99–122.
W. Feit, On integral representations of finite groups. Proc. London Math Soc. (3) 29 (1974), 633–683.
W. Feit, On finite linear groups in dimension at most 10. 397–407 in Proceedings Conference on Finite Groups (ed W. R. Scott and F. Gross), New York Academic Press 1976.
D. F. Holt, W. Plesken, Perfect Groups. Oxford University Press 1989.
D. F. Holt, W. Plesken, A cohomological criterion for a finitely presented group to be infinite, submitted 1990.
C. Jordan, Mémoire sur l’équivalence des formes. J. Ecole Polytech. 48 (1880), 112–150. Oeuvres de C. Jordan, Vol.III, Gauthier-Villars, Paris, 1962, 421–460.
M. Kneser, Zur Theorie der Kristallgitter. Math. Ann. 127, 105–106 (1954).
V. Landazuri, G.M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups. Journal of Algebra 32, 418–443 (1974).
W. Plesken, M. Pohst, On maximal finite irreducible subgroups of GL(n, Z). I. The five-and seven-dimensional case. II. The six-dimensional case, Math. Comp. 31 (138)(1977), 536–577; III. The nine-dimensional case. IV. Remarks on even dimensions with applications to n = 8. V. The eight-dimensional case and a complete description of dimensions less than ten, Math. Comp. 34 (149) (1980), 245–301.
W. Plesken On reducible and decomposable representations of orders. J. reine angew. Math. 297 (1978), 188–210.
W. Plesken, Bravais groups in low dimensions. Proceedings of a Conference on Crystallographic Groups, Bielefeld 1979. match 10 (1981), 97–119.
W. Plesken, Finite unimodular groups of prime degree and circulants. J. of Algebra vol. 97, no. 1 (1985), 286–312.
W. Plesken, Towards a Soluble Qutient Algorithm. J. Symbolic Computation (1987) 4, 111–122.
W. Plesken, W. Hanrath, The lattices of six-dimensional euclidean space. Math, of Comput. vol. 43, no. 168 (1984), 573–587.
S.S. Ryskov, On maximal finite groups of integer n × n-matrices. Dokl. Akad. Nauk SSSR 204 (1972), 561–564. Sov. Math. Dokl. 13 (1972), 720–724.
S.S. Ryskov, Maximal finite groups of integral n × n matrices and full groups of integral automorphisms of positive quadratic forms (Bravais models). Tr. Mat. Inst. Steklov 128 (1972), 183–211. Proc. Steklov Inst. Math. 128 (1972), 217–250.
S.S. Ryskov, Z.D. Lomakina, Proof of a theorem on maximal finite groups of integral 5 × 5-matrices. Proc. Steklov Inst. Math. (1982) Issue 1, 225–235.
W. Scharlau, Quadratic and Hermitian forms. Springer-Verlag 1985.
B. Souvignier, Irreduzible Bravaisgruppen. Diplomarbeit Aachen 1991.
H. Zassenhaus, Neuer Beweis der Endlichkeit der Klassenzahl bei unimodularer Äquivalenz endlicher ganzzahliger Substitutionsgruppen. Hamb. Abh. 12 (1938), 276–288. Additional References for Chapter III
Additional References for Chapter III
L. Auslander, M. Kuranishi, On the holonomy group of locally Euclidean spaces. Ann. Math. 65, 411–415 (1957).
L.S. Charlap, Bieberbach groups and flat manifolds. Berlin Heidelberg New Zork: Springer 1986.
G. Cliff, A. Weiss, Torsion free space groups and permutation lattices for finite groups. Contemporary Mathem. vol 93 (Representation Theory, Group Rings, and Coding Thery) AMS (1989), 123–132.
G. Hiss, A. Szczepanski, On torsion free crystallographic groups. submitted 1990.
H. Hiller, C.-H. Sah, Holonomy of flat manifolds with b 1 = 0. Q.J. Math., Oxf. II Ser. 37, 177–187 (1986).
J. Meyer, Minimal extensions of finite groups by free abelian groups. Arch. Math. 42, 16–31 (1984).
W. Plesken, Minimal dimensions for flat manifolds with prescribed holonomy. Math. Ann. 284, 477–486 (1989).
A. Szczepanski, Five dimensional Bieberbach groups with trivial centre. submitted 1990.
J.A. Wolf, Spaces of constant curvature. New York: McGraw-Hill 1967.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer Basel AG
About this chapter
Cite this chapter
Plesken, W. (1991). Some applications of representation theory. In: Michler, G.O., Ringel, C.M. (eds) Representation Theory of Finite Groups and Finite-Dimensional Algebras. Progress in Mathematics, vol 95. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8658-1_22
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8658-1_22
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9720-4
Online ISBN: 978-3-0348-8658-1
eBook Packages: Springer Book Archive