Nilpotent injectors in Finite Groups All of Whose Local Subgroups Are M-Constrained

Nilpotent injectors have the property that they contain every nilpotent subgroup that they normalize. A. Bialostocki has conjectured [4] that if G is a finite group then all nilpotent injectors of G are conjugate. He has verified his conjecture for the symmetric and alternating groups. The conjecture is also true for soluble groups. The only property of soluble groups used in proving this is C(F(G)) < F(G). So define a group G to be N-constrained if C(F( G)) < F(G). Soluble groups are M-constrained. The definition of nilpotent injector used in this paper is due to Bialostocki and is not the same as the more usual definition given in [ 111. As Bialostocki points out in [4], if G is M-constrained then the two definitions yield the same class of subgroups. The reader is referred to the introduction of [4] for the reason why Bialostocki’s definition is preferable in the context of proving conjugacy theorems. The main theorem of this paper is the following