A bound for the derived and Frattini subgroups of a prime-power group

Ellis, Graham

doi:10.1090/S0002-9939-98-04440-2

A bound for the derived and Frattini subgroups of a prime-power group
HTML articles powered by AMS MathViewer

by Graham Ellis PDF

Proc. Amer. Math. Soc. 126 (1998), 2513-2523 Request permission

Abstract:

This paper is based on the seemingly new observation that the Schur multiplier $M(G)$ of a $d$-generator group of prime-power order $p^n$ has order $|M(G)|\le p^{d(2n-d-1)/2}$. We prove several related results, including sufficient conditions for a sharper bound on $|M(G)|$ to be an equality.

References

Ronald Brown and Jean-Louis Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), no. 3, 311–335. With an appendix by M. Zisman. MR 899052, DOI 10.1016/0040-9383(87)90004-8
J. Burns, G. Ellis, D. MacHale, P. Ó′Murchú, R. Sheehy, and J. Wiegold, Lower central series of groups with small upper central factors, Proc. Royal Irish Acad. (to appear).
Daniel Conduché and Celso Rodríguez-Fernández, Nonabelian tensor and exterior products modulo $q$ and universal $q$-central relative extension, J. Pure Appl. Algebra 78 (1992), no. 2, 139–160. MR 1161338, DOI 10.1016/0022-4049(92)90092-T
Graham J. Ellis, Tensor products and $q$-crossed modules, J. London Math. Soc. (2) 51 (1995), no. 2, 243–258. MR 1325569, DOI 10.1112/jlms/51.2.243
G. J. Ellis and C. Rodríguez-Fernández, An exterior product for the homology of groups with integral coefficients modulo $p$, Cahiers Topologie Géom. Différentielle Catég. 30 (1989), no. 4, 339–343 (English, with French summary). MR 1036346
W. Gaschütz, J. Neubüser, and Ti Yen, Über den Multiplikator von $p$-Gruppen, Math. Z. 100 (1967), 93–96 (German). MR 217181, DOI 10.1007/BF01110785
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Graham Higman, Enumerating $p$-groups. I. Inequalities, Proc. London Math. Soc. (3) 10 (1960), 24–30. MR 113948, DOI 10.1112/plms/s3-10.1.24
Michael R. Jones, Multiplicators of $p$-groups, Math. Z. 127 (1972), 165–166. MR 318304, DOI 10.1007/BF01112608
Michael R. Jones, Some inequalities for the multiplicator of a finite group, Proc. Amer. Math. Soc. 39 (1973), 450–456. MR 314975, DOI 10.1090/S0002-9939-1973-0314975-6
Michael R. Jones, Some inequalities for the multiplicator of a finite group. II, Proc. Amer. Math. Soc. 45 (1974), 167–172. MR 352254, DOI 10.1090/S0002-9939-1974-0352254-2
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory, Second revised edition, Dover Publications, Inc., New York, 1976. Presentations of groups in terms of generators and relations. MR 0422434
I. Schur, Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. reine angew. Math. 127 (1904), 20–50.
Charles C. Sims, Enumerating $p$-groups, Proc. London Math. Soc. (3) 15 (1965), 151–166. MR 169921, DOI 10.1112/plms/s3-15.1.151
Michio Suzuki, Group theory. II, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 248, Springer-Verlag, New York, 1986. Translated from the Japanese. MR 815926, DOI 10.1007/978-3-642-86885-6
M. R. Vaughan-Lee, Breadth and commutator subgroups of $p$-groups, J. Algebra 32 (1974), 278–285. MR 364436, DOI 10.1016/0021-8693(74)90138-0
James Wiegold, Multiplicators and groups with finite central factor-groups, Math. Z. 89 (1965), 345–347. MR 179262, DOI 10.1007/BF01112166