On the relation between upper central quotients and lower central series of a group
HTML articles powered by AMS MathViewer
- by Graham Ellis
- Trans. Amer. Math. Soc. 353 (2001), 4219-4234
- DOI: https://doi.org/10.1090/S0002-9947-01-02812-4
- Published electronically: June 6, 2001
- PDF | Request permission
Abstract:
Let $H$ be a group with a normal subgroup $N$ contained in the upper central subgroup $Z_cH$. In this article we study the influence of the quotient group $G=H/N$ on the lower central subgroup $\gamma _{c+1}H$. In particular, for any finite group $G$ we give bounds on the order and exponent of $\gamma _{c+1}H$. For $G$ equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as $\gamma _{c+1}H$. Our proofs involve: (i) the Baer invariants of $G$, (ii) the Schur multiplier $\mathcal {M}(L,G)$ of $G$ relative to a normal subgroup $L$, and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.References
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- Michael R. Bacon and Luise-Charlotte Kappe, The nonabelian tensor square of a $2$-generator $p$-group of class $2$, Arch. Math. (Basel) 61 (1993), no. 6, 508–516. MR 1254062, DOI 10.1007/BF01196588
- F. Rudolf Beyl and Jürgen Tappe, Group extensions, representations, and the Schur multiplicator, Lecture Notes in Mathematics, vol. 958, Springer-Verlag, Berlin-New York, 1982. MR 681287
- R. Brown, D. L. Johnson, and E. F. Robertson, Some computations of nonabelian tensor products of groups, J. Algebra 111 (1987), no. 1, 177–202. MR 913203, DOI 10.1016/0021-8693(87)90248-1
- 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
- John Burns and Graham Ellis, On the nilpotent multipliers of a group, Math. Z. 226 (1997), no. 3, 405–428. MR 1483540, DOI 10.1007/PL00004348
- John Burns and Graham Ellis, Inequalities for Baer invariants of finite groups, Canad. Math. Bull. 41 (1998), no. 4, 385–391. MR 1658215, DOI 10.4153/CMB-1998-051-3
- 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. Roy. Irish Acad. Sect. A 97 (1997), no. 2, 113–122. MR 1645267
- Graham Ellis, On groups with a finite nilpotent upper central quotient, Arch. Math. (Basel) 70 (1998), no. 2, 89–96. MR 1491453, DOI 10.1007/s000130050169
- Graham Ellis, The Schur multiplier of a pair of groups, Appl. Categ. Structures 6 (1998), no. 3, 355–371. MR 1641859, DOI 10.1023/A:1008652316165
- Graham Ellis, A bound for the derived and Frattini subgroups of a prime-power group, Proc. Amer. Math. Soc. 126 (1998), no. 9, 2513–2523. MR 1459119, DOI 10.1090/S0002-9939-98-04440-2
- Graham Ellis, On the computation of certain homotopical-functors, LMS J. Comput. Math. 1 (1998), 25–41. MR 1635723, DOI 10.1112/S1461157000000139
- Graham Ellis and Aidan McDermott, Tensor products of prime-power groups, J. Pure Appl. Algebra 132 (1998), no. 2, 119–128. MR 1640071, DOI 10.1016/S0022-4049(97)00112-6
- Frölich, A.: Baer invariants of algebras. Trans. Amer. Math. Soc. 109, 221-244 (1962)
- 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
- N. D. Gupta and M. R. R. Moghaddam, Higher Schur-multiplicators of nilpotent dihedral groups, C. R. Math. Rep. Acad. Sci. Canada 14 (1992), no. 5, 225–230. MR 1199865
- 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
- Philip Hall, The Edmonton notes on nilpotent groups, Queen Mary College Mathematics Notes, Queen Mary College, Mathematics Department, London, 1969. MR 0283083
- 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
- Jean-Louis Loday, Cohomologie et groupe de Steinberg relatifs, J. Algebra 54 (1978), no. 1, 178–202 (French). MR 511461, DOI 10.1016/0021-8693(78)90025-X
- Alexander Lubotzky and Avinoam Mann, Powerful $p$-groups. I. Finite groups, J. Algebra 105 (1987), no. 2, 484–505. MR 873681, DOI 10.1016/0021-8693(87)90211-0
- Abraham S.-T. Lue, The Ganea map for nilpotent groups, J. London Math. Soc. (2) 14 (1976), no. 2, 309–312. MR 430103, DOI 10.1112/jlms/s2-14.2.309
- John L. MacDonald, Group derived functors, J. Algebra 10 (1968), 448–477. MR 246936, DOI 10.1016/0021-8693(68)90072-0
- D. MacHale and P. Ó Murchú, Commutator subgroups of groups with small central factor groups, Proc. Roy. Irish Acad. Sect. A 93 (1993), no. 1, 123–129. MR 1241846
- 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
- M. R. R. Moghaddam and B. Mashayekhy, Higher Schur-multiplicator of a finite abelian group, Algebra Colloq. 4 (1997), no. 3, 317–322. MR 1681548
- James Wiegold, Multiplicators and groups with finite central factor-groups, Math. Z. 89 (1965), 345–347. MR 179262, DOI 10.1007/BF01112166
- James Wiegold, Commutator subgroups of finite $p$-groups, J. Austral. Math. Soc. 10 (1969), 480–484. MR 0258961
Bibliographic Information
- Graham Ellis
- Affiliation: Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, Bonn, Germany
- Address at time of publication: Department of Mathematics, National University of Ireland, Galway, Ireland
- Email: graham.ellis@nuigalway.ie
- Received by editor(s): February 12, 1999
- Published electronically: June 6, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 4219-4234
- MSC (2000): Primary 20F14, 20F12
- DOI: https://doi.org/10.1090/S0002-9947-01-02812-4
- MathSciNet review: 1837229