On the double frobenius groups and their characters
Abstract
The Double Frobenius group is the result of the action of a Frobenius group H = NH, with
kernel N and complement H, on a finite group G. If the action of H on G is such that, N acts
fixed point free on G and GN is also a Frobenius group with kernel G and complement N, then
G = GNH = G: (N:H) = (G:N ):H is a double Frobenius group. In this study we briefly describe the
structure of the double Frobenius group and then construct in general two double Frobenius groups
which have the form 2n:(Z2n-1:Zn), where n is a prime such that 2n - 1 is a Mersenne prime and
22r:(Z2r_1:Z2), where 2 :S r EN respectively. We then proceed to analyse the two double Frobenius
groups mentioned above, calculating the conjugacy classes, Fischer matrices and character table of
the groups. The study is concluded by demonstrating these calculations of the conjugacy classes,
Fischer matrices and character tables of two examples of each type of double Frobenius group,
namely, 23:(Z7:Z3) and 25 :(Z31 :Zs) for the type 2n:(Z2n_ 1 :Zn) with n = 3 and n = 5 respectively,
and 24:(Z3:Z2) and 26:(Z7:Z2) for the type 22r:(Z2r-1 :Z2) with r = 2 and r = 3 respectively