×

The Whitehead group of (almost) extra-special \(p\)-groups with \(p\) odd. (English) Zbl 1403.19003

Authors’ abstract: Let \(p\) be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special \(p\)-groups. For this we compute, for any finite \(p\)-group \(P\), the subgroup \(C l_1(\mathbb{Z} P)\) of \(S K_1(\mathbb{Z} P)\), in terms of a genetic basis of \(P\). We also introduce a deflation map \(C l_1(\mathbb{Z} P) \rightarrow C l_1(\mathbb{Z}(P / N))\), for a normal subgroup \(N\) of \(P\), and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of \(S K_1(\mathbb{Z} P)\), when \(P\) is an elementary abelian \(p\)-group.

MSC:

19B28 \(K_1\) of group rings and orders
20D15 Finite nilpotent groups, \(p\)-groups
20C05 Group rings of finite groups and their modules (group-theoretic aspects)

Software:

GAP
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alperin, R. C.; Dennis, R. K.; Oliver, R.; Stein, M. R., \(S K_1\) of finite abelian groups, II, Invent. Math., 87, 253-302, (1987) · Zbl 0605.18006
[2] Barker, L., Genotypes of irreducible representations of finite p-groups, J. Algebra, 306, 655-681, (2007) · Zbl 1139.19002
[3] Benson, D. J., Representations and cohomology II, Cambridge Studies in Advanced Mathematics, vol. 31, (1991), Cambridge University Press · Zbl 0731.20001
[4] Bouc, S., K-theory, genotypes, and p-biset functors, (2016), preprint
[5] Bouc, S., The Dade group of a p-group, Invent. Math., 164, 189-231, (2006) · Zbl 1099.20004
[6] Bouc, S., The functor of units of Burnside rings for p-groups, Comment. Math. Helv., 82, 583-615, (2007) · Zbl 1142.19001
[7] Bouc, S., Biset functors for finite groups, (2010), Springer Berlin · Zbl 1205.19002
[8] Bouc, S., Fast decomposition of p-groups in the roquette category, for \(p > 2\), (RIMS Kôkyûroku, vol. 1872, (2014)), 113-121
[9] Bouc, S.; Mazza, N., The Dade group of (almost) extraspecial p-groups, J. Pure Appl. Algebra, 192, 21-51, (2004) · Zbl 1055.20008
[10] Carlson, J. F.; Thévenaz, J., Torsion endo-trivial modules, Algebr. Represent. Theory, 3, 303-335, (2000) · Zbl 0970.20004
[11] Doerk, K.; Hawkes, T., Finite soluble groups, (1992), Walter de Gruyter Berlin · Zbl 0753.20001
[12] Huppert, B., Endliche gruppen I, Die Grundlehren der mathematischen Wissenschaften, vol. 134, (1967), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0217.07201
[13] GAP - groups, algorithms, and programming, version 4.8.5, (2016)
[14] Gorenstein, D., Finite groups, (1968), Chelsea Publishing New York · Zbl 0185.05701
[15] Guaschi, J.; Juan-Pineda, D.; Millán López, S., The lower algebraic K-theory of the braid groups of the sphere, (2012), preprint
[16] Lafont, J.-F.; Magurn, B. A.; Ortiz, I. J., Lower algebraic K-theory of certain reflection groups, Proc. Camb. Philos. Soc., 148, 193-226, (2010) · Zbl 1190.19001
[17] Oliver, R., Whitehead groups of finite groups, (1988), Cambridge University Press UK · Zbl 0636.18001
[18] Romero, N., Computing Whitehead groups using genetic bases, J. Algebra, 450, 646-666, (2016) · Zbl 1401.19004
[19] Roquette, P., Realisierung von darstellungen endlicher nilpotenter gruppen, Arch. Math., 9, 241-250, (1958) · Zbl 0083.25002
[20] Serre, J.-P., Linear representations of finite groups, (1977), Springer-Verlag New York · Zbl 0355.20006
[21] Ushitaki, F., \(S K_1(\mathbb{Z} [G])\) of finite solvable groups which act linearly and freely on spheres, Osaka J. Math., 28, 1, 117-127, (1991) · Zbl 0741.19002
[22] Ushitaki, F., A generalization of a theorem of Milnor, Osaka J. Math., 31, 403-415, (1994) · Zbl 0853.57032
[23] Whitehead, J. H.C., Simple homotopy types, Am. J. Math., 72, 1-57, (1950) · Zbl 0040.38901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.