Presentations for unit groups of modular group algebras of groups of order 16

Sandling, Robert

doi:10.1090/S0025-5718-1992-1136226-5

Presentations for unit groups of modular group algebras of groups of order $16$
HTML articles powered by AMS MathViewer

by Robert Sandling PDF

Math. Comp. 59 (1992), 689-701 Request permission

Abstract:

For a p-group G and the field F of p elements, let V denote the group of normalized units of the group algebra FG. Generators and relations are provided for V for each group G of order dividing 16. The presentations are sufficiently concise to permit transcription for machine calculation with V. Some applications are described. A theoretical method for obtaining presentations for V is developed. It is most effective when $p = 2$, or when $p = 3$ and the commutator subgroup G’ is of order 3. Implementation involves extensive calculation in FG.

References

Czesław Bagiński, Groups of units of modular group algebras, Proc. Amer. Math. Soc. 101 (1987), no. 4, 619–624. MR 911020, DOI 10.1090/S0002-9939-1987-0911020-6
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
John J. Cannon, An introduction to the group theory language, Cayley, Computational group theory (Durham, 1982) Academic Press, London, 1984, pp. 145–183. MR 760656
R. Keith Dennis, The structure of the unit group of group rings, Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Okla., 1975) Lecture Notes in Pure and Appl. Math., Vol. 26, Dekker, New York, 1977, pp. 103–130. MR 0444697
Narain Gupta and Frank Levin, On the Lie ideals of a ring, J. Algebra 81 (1983), no. 1, 225–231. MR 696135, DOI 10.1016/0021-8693(83)90217-X
Marshall Hall Jr. and James K. Senior, The groups of order $2^{n}\,(n\leq 6)$, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1964. MR 0168631
Lee R. Ivory, A note on normal complements in mod $p$ envelopes, Proc. Amer. Math. Soc. 79 (1980), no. 1, 9–12. MR 560574, DOI 10.1090/S0002-9939-1980-0560574-4
S. A. Jennings, The structure of the group ring of a $p$-group over a modular field, Trans. Amer. Math. Soc. 50 (1941), 175–185. MR 4626, DOI 10.1090/S0002-9947-1941-0004626-6
D. L. Johnson, Presentations of groups, London Mathematical Society Student Texts, vol. 15, Cambridge University Press, Cambridge, 1990. MR 1056695
R. Laue, J. Neubüser, and U. Schoenwaelder, Algorithms for finite soluble groups and the SOGOS system, Computational group theory (Durham, 1982) Academic Press, London, 1984, pp. 105–135. MR 760654
Avinoam Mann and Aner Shalev, The nilpotency class of the unit group of a modular group algebra. II, Israel J. Math. 70 (1990), no. 3, 267–277. MR 1074492, DOI 10.1007/BF02801464
L. E. Moran and R. N. Tench, Normal complements in $\textrm {mod}\ p$-envelopes, Israel J. Math. 27 (1977), no. 3-4, 331–338. MR 447403, DOI 10.1007/BF02756491

GAP

Robert Sandling, Units in the modular group algebra of a finite abelian $p$-group, J. Pure Appl. Algebra 33 (1984), no. 3, 337–346. MR 761637, DOI 10.1016/0022-4049(84)90066-5
Robert Sandling, The isomorphism problem for group rings: a survey, Orders and their applications (Oberwolfach, 1984) Lecture Notes in Math., vol. 1142, Springer, Berlin, 1985, pp. 256–288. MR 812504, DOI 10.1007/BFb0074806
Robert Sandling, The modular group algebra of a central-elementary-by-abelian $p$-group, Arch. Math. (Basel) 52 (1989), no. 1, 22–27. MR 980047, DOI 10.1007/BF01197966
Aner Shalev, On the conjectures concerning units in $p$-group algebras, Proceedings of the Second International Group Theory Conference (Bressanone, 1989), 1990, pp. 279–288. MR 1068368
Aner Shalev, The nilpotency class of the unit group of a modular group algebra. I, Israel J. Math. 70 (1990), no. 3, 257–266. MR 1074491, DOI 10.1007/BF02801463

Group tables

C. T. C. Wall, Norms of units in group rings, Proc. London Math. Soc. (3) 29 (1974), 593–632. MR 376746, DOI 10.1112/plms/s3-29.4.593

Some results on the group algebra of a group over a prime field

Ein Verfahren, jeder endlichen p-Gruppe einen Lie-Ring mit der Charakteristik p zuzuordnen

13