Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The modular group algebra problem for metacyclic $p$-groups


Author: Robert Sandling
Journal: Proc. Amer. Math. Soc. 124 (1996), 1347-1350
MSC (1991): Primary 20C05; Secondary 16S34, 16U60, 20C20, 20D15, 20F05
DOI: https://doi.org/10.1090/S0002-9939-96-03518-6
MathSciNet review: 1343723
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the isomorphism type of a metacyclic $p$-group is determined by its group algebra over the field $F$ of $p$ elements. This completes work of Baginski. It is also shown that, if a $p$-group $G$ has a cyclic commutator subgroup $G'$, then the order of the largest cyclic subgroup containing $G'$ is determined by $FG$.


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

  • 1. C. Baginski, The isomorphism question for modular group algebras of metacyclic $p$-groups, Proc. Amer. Math. Soc. 104 (1988), 39--42. MR 89i:20016
  • 2. F. R. Beyl, The classification of metacyclic $p$-groups, and other applications of homological algebra to group theory, Ph. D. thesis, Cornell Univ., 1972.
  • 3. B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967. MR 37:302
  • 4. B. W. King, Presentations of metacyclic groups, Bull. Austral. Math. Soc. 8 (1973), 103--131. MR 48:2246
  • 5. B. Külshammer, Bemerkungen über die Gruppenalgebra als symmetrische Algebra, II, J. Algebra 75 (1982), 59--69. MR 83j:16017b
  • 6. ------, Group-theoretical descriptions of ring-theoretical invariants of group algebras, Prog. in Math. 95 (1991), 425--442. MR 92d:16037
  • 7. S. Liedahl, Presentations of metacyclic $p$-groups with applications to $K$-admissibility questions, J. Algebra 169 (1994), 965--983. CMP 95:03
  • 8. M. F. Newman and M. Xu, Metacyclic groups of prime-power order, Adv. Math. (Beijing) 17 (1988), 106--107.
  • 9. L. Redéi, Endliche $p$-Gruppen, Akadémiai Kiadó, Budapest, 1989. MR 90i:20001
  • 10. R. Sandling, The isomorphism problem for group rings: a survey, Orders and their applications (Oberwolfach, 1984), 256--288, Lecture Notes in Mathematics 1142, Springer, Berlin, 1985. MR 87b:20007
  • 11. ------, The modular group algebra of a central-elementary-by-abelian $p$-group, Arch. Math. (Basel) 52 (1989), 22--27. MR 90b:20007
  • 12. A. Shalev, Dimension subgroups, nilpotency indices, and the number of generators of ideals, J. Algebra 129 (1990), 412--438. MR 91a:20011
  • 13. M. Wursthorn, Die modularen Gruppenringe der Gruppen der Ordnung $2^6$, Diplomarbeit, Universität Stuttgart, 1990.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 20C05, 16S34, 16U60, 20C20, 20D15, 20F05

Retrieve articles in all journals with MSC (1991): 20C05, 16S34, 16U60, 20C20, 20D15, 20F05


Additional Information

Robert Sandling
Affiliation: Department of Mathematics, The University, Manchester M13 9PL, England
Email: rsandling@manchester.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-96-03518-6
Keywords: Modular group algebra, $p$-group, isomorphism problem, metacyclic
Received by editor(s): July 11, 1994
Communicated by: Ronald M. Solomon
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society