Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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


Author: Robert Sandling
Journal: Math. Comp. 59 (1992), 689-701
MSC: Primary 16S34; Secondary 16U60, 20C05, 20D15, 20F05
DOI: https://doi.org/10.1090/S0025-5718-1992-1136226-5
MathSciNet review: 1136226
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] C. Baginski, Groups of units of modular group algebras, Proc. Amer. Math. Soc. 101 (1987), 619-624. MR 911020 (88j:16015)
  • [2] H. Bass, Algebraic K-theory, Benjamin, New York, 1968. MR 0249491 (40:2736)
  • [3] J. J. Cannon, An introduction to the group theory language, Cayley, Computational Group Theory (M. D. Atkinson, ed.), Academic Press, London, 1984, pp. 145-183. MR 760656
  • [4] R. K. Dennis, The structure of the unit group of group rings, Ring Theory II (B. R. McDonald and R. A. Morris, eds.), Dekker, New York, 1977, pp. 103-130. MR 0444697 (56:3047)
  • [5] N. Gupta and F. Levin, On the Lie ideals of a ring, J. Algebra 81 (1983), 225-231. MR 696135 (84i:16036)
  • [6] M. Hall and J. Senior, The groups of order $ {2^n}\;(n \leq 6)$, Macmillan, New York, 1964. MR 0168631 (29:5889)
  • [7] L. R. Ivory, A note on normal complements in $ \bmod\, p$ envelopes, Proc. Amer. Math. Soc. 79 (1980), 9-12. MR 560574 (82e:20004)
  • [8] 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 0004626 (3:34f)
  • [9] D. L. Johnson, Presentations of groups, Cambridge Univ. Press, Cambridge, 1990. MR 1056695 (91h:20001)
  • [10] R. Laue, J. Neubüser, and U. Schoenwaelder, Algorithms for finite soluble groups and the SOGOS system, Computational Group Theory (M. D. Atkinson, ed.), Academic Press, London, 1984, pp. 105-135. MR 760654 (86h:20023)
  • [11] A. Mann and A. Shalev, The nilpotency class of the unit group of a modular group algebra. II, Israel J. Math. 70 (1990), 267-277. MR 1074492 (92a:16030)
  • [12] L. E. Moran and R. N. Tench, Normal complements in $ \bmod\, p$-envelopes, Israel J. Math. 27 (1977), 331-338. MR 0447403 (56:5715)
  • [13] A. Niemeyer, W. Nickel, and M. Schönert, GAP, Lehrstuhl D, Mathematik., Rhein.-Westf. Techn. Hochschule, Aachen, 1988.
  • [14] R. Sandling, Units in the modular group algebra of a finite abelian p-group, J. Pure Appl. Algebra 33 (1984), 337-346. MR 761637 (86a:16012)
  • [15] -, 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 (87b:20007)
  • [16] -, The modular group algebra of a central-elementary-by-abelian p-group, Arch. Math. (Basel) 52 (1989), 22-27. MR 980047 (90b:20007)
  • [17] A. Shalev, On some conjectures concerning units in p-group algebras, Proc. 2nd Internat. Group Theory Conf. (Bressanone, 1989). Rend. Circ. Mat. Palermo (2) Suppl. 23 (1990), 279-288. MR 1068368 (91h:20009)
  • [18] -, The nilpotency class of the unit group of a modular group algebra. I, Israel J. Math. 70 (1990), 257-266. MR 1074491 (92a:16029)
  • [19] A. D. Thomas and G. V. Wood, Group tables, Shiva, Nantwich, U.K., 1980.
  • [20] C. T. C. Wall, Norms of units in group rings, Proc. London Math. Soc. (3) 29 (1974), 593-632. MR 0376746 (51:12921)
  • [21] H. N. Ward, Some results on the group algebra of a group over a prime field, Seminar on Finite Groups and Related Topics, Mimeographed notes, Harvard Univ., 1960-1961, pp. 13-19.
  • [22] H. Zassenhaus, Ein Verfahren, jeder endlichen p-Gruppe einen Lie-Ring mit der Charakteristik p zuzuordnen, Abh. Math. Sem. Hansische Univ. 13 (1939), 200-207.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 16S34, 16U60, 20C05, 20D15, 20F05

Retrieve articles in all journals with MSC: 16S34, 16U60, 20C05, 20D15, 20F05


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1992-1136226-5
Keywords: Modular group algebra, p-group, unit group, generators and relations, presentation of an extension
Article copyright: © Copyright 1992 American Mathematical Society

American Mathematical Society