ON THE DOUBLE FROBENIUS GROUP OF THE FORM \(2^{2r}\):(\(\mathbb{Z}_{2^{r}-1}:\mathbb{Z}_2\))

Abstract

Let G be a finite group.  Let \(\overline{H}\) = NH be a Frobenius group with kernel N and complement H.  If G admits \(\overline{H}\) as a group of automorphisms such that C_G(N) = {1_G} and GN is also a Frobenius group with kernel G and complement N, then \(\overline{G}\) = GNH is called a double Frobenius group (or 2-Frobenius group).  The group \(\overline{G}\) = GNH is a product of subgroups G ≤ \(\overline{G}\), N ≤ \(\overline{G}\), H ≤ \(\overline{G}\) with G \(\trianglelefteq\) \(\overline{G}\), GN \(\trianglelefteq\) \(\overline{G}\) and \(\overline{G}\) = G:NH = GN:H.  In this article we shall construct a double Frobenius group of the form \(\overline{G}\) = \(2^{2r}\):(\(\mathbb{Z}_{2^{r}-1}:\mathbb{Z}_2\)), where G \(\cong\) \(2^{2r}\), N \(\cong\) \(\mathbb{Z}_{2^{r}-1}\) and H \(\cong\) \(\mathbb{Z}_2\), where r ∈ \(\mathbb{N}\), r ≥ 2.  The construction is a general one that gives examples of double Frobenius groups for particular values of n.  In addition to the general construction of the group \(\overline{G}\) = \(2^{2r}\):(\(\mathbb{Z}_{2^{r}-1}:\mathbb{Z}_2\)), we calculate in general the conjugacy classes, Fischer matrices and character table of the group.  One example \(\overline{G}\) = \(2^{4}\):(\(\mathbb{Z}_{3}:\mathbb{Z}_2\)), (the case r = 2) is demonstrated.