On non-solvable Camina pairs

In this paper we study non-solvable and non-Frobenius Camina pairs (G, N). It is known [D. Chillag, A. Mann, C. Scoppola, Generalized Frobenius groups II, Israel J. Math. 62 (1988) 269-282] that in this case N is a p-group. Our first result (Theorem 1.3) shows that the solvable residual of G / O_p (G) is isomorphic either to SL (2, p^e), p is a prime or to SL (2, 5), SL (2, 13) with p = 3, or to SL (2, 5) with p ≥ 7. Our second result provides an example of a non-solvable and non-Frobenius Camina pair (G, N) with | O_p (G) | = 5⁵ and G / O_p (G) ≅ SL (2, 5). Note that G has a character which is zero everywhere except on two conjugacy classes. Groups of this type were studies by S.M. Gagola [S.M. Gagola, Characters vanishing on all but two conjugacy classes, Pacific J. Math. 109 (1983) 363-385]. To our knowledge this group is the first example of a Gagola group which is non-solvable and non-Frobenius.