Czechoslovak Mathematical Journal, Vol. 73, No. 1, pp. 71-100, 2023


On some finite 2-groups in which the derived group has two generators

Elliot Benjamin, Chip Snyder

Received November 1, 2021.   Published online June 20, 2022.

Abstract:  We show that any finite 2-group, whose abelianization has either 4-rank at most 2 or 8-rank 0 and whose commutator subgroup is generated by two elements, is metabelian. We also prove that the minimal order of any 2-group with nonabelian commutator subgroup of 2-rank 2 is $2^{12}$.
Keywords:  2-group; metabelian
Classification MSC:  20D15


References:
[1] E. Benjamin, C. Snyder: Some real quadratic number fields whose Hilbert 2-class fields have class number congruent to 2 modulo 4. Acta Arith. 177 (2017), 375-392. DOI 10.4064/aa8485-9-2016 | MR 3630722 | Zbl 1401.11143
[2] N. Blackburn: On prime-power groups in which the derived group has two generators. Proc. Camb. Philos. Soc. 53 (1957), 19-27. DOI 10.1017/S0305004100031959 | MR 0081904 | Zbl 0077.03202
[3] N. Blackburn: On prime-power groups with two generators. Proc. Camb. Philos. Soc. 54 (1958), 327-337. DOI 10.1017/S0305004100033521 | MR 0102557 | Zbl 0083.01902
[4] B. Huppert: Endliche Gruppen. I. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen 134. Springer, Berlin (1967). (In German.) DOI 10.1007/978-3-642-64981-3 | MR 0224703 | Zbl 0217.07201

Affiliations:   Elliot Benjamin (corresponding author), School of Social and Behavioral Sciences, Capella University, 225 South 6th Street, Minneapolis, MN 55402, USA, e-mail: ben496@prexar.com; Chip Snyder, Department of Mathematics and Statistics, University of Maine, 333 Neville Hall, Orono, ME 04469-5752, USA, e-mail: wsnyder@maine.edu


 
PDF available at: