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