The modular group algebra problem for metacyclic 𝑝-groups

Sandling, Robert

doi:10.1090/S0002-9939-96-03518-6

The modular group algebra problem for metacyclic $p$-groups
HTML articles powered by AMS MathViewer

by Robert Sandling PDF

Proc. Amer. Math. Soc. 124 (1996), 1347-1350 Request permission

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 Bagiński. 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

Czesław Bagiński, The isomorphism question for modular group algebras of metacyclic $p$-groups, Proc. Amer. Math. Soc. 104 (1988), no. 1, 39–42. MR 958039, DOI 10.1090/S0002-9939-1988-0958039-8
F. R. Beyl, The classification of metacyclic $p$-groups, and other applications of homological algebra to group theory, Ph. D. thesis, Cornell Univ., 1972.
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
Bruce W. King, Presentations of metacyclic groups, Bull. Austral. Math. Soc. 8 (1973), 103–131. MR 323893, DOI 10.1017/S0004972700045500
Burkhard Külshammer, Bemerkungen über die Gruppenalgebra als symmetrische Algebra, J. Algebra 72 (1981), no. 1, 1–7 (German). MR 634613, DOI 10.1016/0021-8693(81)90308-2
Burkhard Külshammer, Group-theoretical descriptions of ring-theoretical invariants of group algebras, Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991) Progr. Math., vol. 95, Birkhäuser, Basel, 1991, pp. 425–442. MR 1112173, DOI 10.1007/978-3-0348-8658-1_{1}9
S. Liedahl, Presentations of metacyclic $p$-groups with applications to $K$-admissibility questions, J. Algebra 169 (1994), 965–983.
M. F. Newman and M. Xu, Metacyclic groups of prime-power order, Adv. Math. (Beijing) 17 (1988), 106–107.
László Rédei, Endliche $p$-Gruppen, Akadémiai Kiadó, Budapest, 1989 (German). MR 992619
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, Dimension subgroups, nilpotency indices, and the number of generators of ideals in $p$-group algebras, J. Algebra 129 (1990), no. 2, 412–438. MR 1040946, DOI 10.1016/0021-8693(90)90228-G
M. Wursthorn, Die modularen Gruppenringe der Gruppen der Ordnung $2^6$, Diplomarbeit, Universität Stuttgart, 1990.