×

Characterization of finite \(p\)-groups by their non-abelian tensor square. (English) Zbl 1276.19002

The tensor product \(G \otimes H\) of two nonabelian groups is defined in terms of symbols by twisted multiplicativity relations, resulting from compatible actions of the groups on each other. The non-abelian tensor square \(G \otimes G\) was computed for some classes of groups, including extra special groups.
It is known that if \(|G|=p^n\) and \(|G'|=p^c\), then \(|G\otimes G|=p^{n(n-c)-\ell}\) for a suitable \(\ell\geq 0\). This paper lists all the \(p\)-groups with \(\ell \leq 10\). It turns out that for every \(p\) (even or odd), there are ten such groups.
The classification uses the fact that for a \(p\)-group \(G\), \(|G \otimes G| = 2^k|G|\cdot |M(G)| \cdot |M(G^{\mathrm{ab}})|\), where \(M(\cdot)\) is the Schur multiplier and the factor \(2^k\), dividing \(|G/G'|\), is present only for \(2\)-groups. The authors then exploit a classification of the \(p\)-groups with \(t(G)\leq 6\) where \(|M(G)|=p^{n(n-1)/2-t(G)}\), and the GAP database of small groups.

MSC:

19C09 Central extensions and Schur multipliers
20D15 Finite nilpotent groups, \(p\)-groups

Software:

GAP
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bacon M. R., Arch. Math. 61 pp 508– (1993) · Zbl 0823.20021 · doi:10.1007/BF01196588
[2] DOI: 10.1016/0021-8693(91)90106-I · Zbl 0739.20005 · doi:10.1016/0021-8693(91)90106-I
[3] DOI: 10.1515/JGT.2009.032 · Zbl 1206.20033 · doi:10.1515/JGT.2009.032
[4] DOI: 10.1016/0021-8693(87)90248-1 · Zbl 0626.20038 · doi:10.1016/0021-8693(87)90248-1
[5] DOI: 10.1016/0040-9383(87)90004-8 · Zbl 0622.55009 · doi:10.1016/0040-9383(87)90004-8
[6] Ellis G., J. Algebra 111 pp 203– (1987) · Zbl 0626.20039 · doi:10.1016/0021-8693(87)90249-3
[7] DOI: 10.1080/00927879908826689 · Zbl 0947.20008 · doi:10.1080/00927879908826689
[8] Ellis G., Bull. Austral. Math. Soc. 60 (2) pp 191– (1999) · Zbl 0940.20017 · doi:10.1017/S0004972700036327
[9] DOI: 10.1016/S0022-4049(97)00112-6 · Zbl 0959.20030 · doi:10.1016/S0022-4049(97)00112-6
[10] The GAP Group, GAP-Groups, Algorithms and Programming, Version 4.4.12, 2008 (http://www.gap-system.org/).
[11] DOI: 10.1098/rspa.1956.0198 · Zbl 0071.02301 · doi:10.1098/rspa.1956.0198
[12] Hannebauer T., Arch. Math. 55 pp 30– (1990) · Zbl 0763.20012 · doi:10.1007/BF01199111
[13] DOI: 10.1080/00927870600936831 · Zbl 1109.20012 · doi:10.1080/00927870600936831
[14] Johnson D. L., Proc. Edinburg Math. Soc. 30 pp 91– (1987) · Zbl 0588.20022 · doi:10.1017/S0013091500018009
[15] Jones M. R., Proc. Amer. Math. Soc. 39 pp 450– (1973) · doi:10.1090/S0002-9939-1973-0314975-6
[16] DOI: 10.1017/S0017089599000014 · Zbl 0970.20012 · doi:10.1017/S0017089599000014
[17] Karpilovsky G., The Schur Multiplier LMS Monogrphs New Series 2 (1987)
[18] Robinson D. J. F., A Course in the Theory of Group., 2. ed. (1996) · doi:10.1007/978-1-4419-8594-1
[19] DOI: 10.1007/BF01244898 · Zbl 0791.20020 · doi:10.1007/BF01244898
[20] DOI: 10.1080/00927879408824951 · Zbl 0819.20032 · doi:10.1080/00927879408824951
[21] DOI: 10.1080/00927870601142132 · Zbl 1120.20020 · doi:10.1080/00927870601142132
[22] Stancu R., J. Algebra 249 pp 120– (2002) · Zbl 1004.20008 · doi:10.1006/jabr.2001.9069
[23] DOI: 10.2307/1969511 · Zbl 0037.26101 · doi:10.2307/1969511
[24] DOI: 10.1080/00927879408824827 · Zbl 0832.20038 · doi:10.1080/00927879408824827
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.