Injectors with a normal complement in a finite solvable group. (Q651236)

Let \(G\) be a finite soluble group and \(H\) a subgroup of \(G\). The main results of the paper are as follows. Theorem 1: If \(S\) a subnormal subgroup of \(H\), \(K\) a normal complement of \(H\) in \(G\) and \(M\) a subgroup of \(K\) that is normal in \(SM\), then the following statements are equivalent: (i) \(H\) is an injector in \(G\) with respect to some Fitting set; (ii) \(S\) is pronormal in \(SK\); (iii) \(M=C_M(S)[M,S]\); (iv) \([M,S]=[M,S,S]\). Theorem 2: If \(X\) is a subgroup of \(G\) that is subnormal in \(\langle H,X\rangle\), the following statements are equivalent: (i) \(H\) is an injector in \(G\) with respect to some Fitting set; (ii) \(N_G(X)\subseteq N_G(H\cap X)X\); (iii) \((H\cap X)X'\) is a normal subgroup of \(N_G(X)\).