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Schur multiplicators of finite \(p\)-groups with fixed coclass. (English) Zbl 1153.20015

The author investigates the Schur multiplicators \(M(G)\) of finite \(p\)-groups \(G\) using the coclass as primary invariant (the coclass of a group of order \(p^n\) and class \(c\) is \(n-c\)). For \(p>2\), she proves that there are at most finitely many \(p\)-groups \(G\) of coclass \(r\) with \(|M(G)|\leq s\) for every \(r\) and \(s\). This is not true for \(p=2\). Infinite series of 2-groups \(G\) with coclass \(r\) and trivial Schur multiplicator are constructed. Coclass theory is used to further investigate the Schur multiplicators of 2-groups of coclass \(r\).

MSC:

20D15 Finite nilpotent groups, \(p\)-groups
20D60 Arithmetic and combinatorial problems involving abstract finite groups

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References:

[1] M. du Sautoy, Counting p-groups and nilpotent groups, Publications Mathématiques. Institut de Hautes Études Scientifiques 92 (2001), 63–112. · Zbl 1017.20012
[2] B. Eick, Schur multiplicators of infinite pro-p-groups with finite coclass, Israel Journal of Mathematics 166 (2008), 147–156. · Zbl 1158.20011 · doi:10.1007/s11856-008-1024-z
[3] B. Eick and C. R. Leedham-Green, On the classification of prime-power groups by coclass, The Bulletin of the London Mathematical Society 40 (2008), 274–288. · Zbl 1168.20007 · doi:10.1112/blms/bdn007
[4] T. Ganea, Homologie et extensions centrales de groupes, Comptes Rendus Mathématique. Académie des Sciences. Paris Sér. A–B 266 (1968), A556–A558. · Zbl 0175.29702
[5] G. Karpilovski, Projective representations of finite groups, Marcel Dekker, INC, 1985.
[6] C. R. Leedham-Green and S. McKay, The Structure of Groups of Prime Power Order, London Mathematical Society Monographs, Oxford Science Publications, 2002. · Zbl 1008.20001
[7] A. Lubotzky and A. Mann, Powerful p-groups. I. Finite groups, Journal of Algebra 105(2) (1987), 484–505. · Zbl 0626.20010 · doi:10.1016/0021-8693(87)90211-0
[8] A. Mann, Some questions about p-group, Journal of the Australian Mathematical Society 67 (1999), 356–379. · Zbl 0944.20012 · doi:10.1017/S1446788700002068
[9] M. F. Newman and E. A. O’Brien, Classifying 2-groups by coclass, Transactions of the American Mathematical Society 351 (1999), 131–169. · Zbl 0914.20020 · doi:10.1090/S0002-9947-99-02124-8
[10] D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, volume 80, Springer-Verlag, New York, Heidelberg, Berlin, 1982. · Zbl 0483.20001
[11] I. Schur, Über die Darstellungen endlicher Gruppen durch gebrochene lineare Substitutionen, Journal of Mathematics 127 (1904), 20–50. · JFM 35.0155.01
[12] The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.4, Available from http://www.gap-system.org , 2005.
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