Portal de Periódicos da CAPES

On non-solvable Camina pairs

2009; Elsevier BV; Volume: 322; Issue: 7 Linguagem: Inglês

10.1016/j.jalgebra.2009.06.023

1090-266X

Zvi Arad, Avinoam Mann, Mikhail Muzychuk, Christian Pech,

Homotopy and Cohomology in Algebraic Topology

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 ) | = 5 5 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.