ABSTRACT
K-monomorphism M→L, so is a K-automorphism of M by Theorem 11.9. Hence τ(M)=M. By Lemma 12.2, τM*τ−1=M*, so M* is a normal subgroup of G.
Conversely, suppose that M* is a normal subgroup of G. Let σ be any K-monomorphism M→L. By Theorem 11.3, there is a K-automorphism τ of L such that τ|M=σ. Now τM*τ
−1=M* since M* is a normal subgroup of G, so by Lemma 12.2, τ(M)*=M*. By part 2 of Theorem 12.1, τ(M)=M. Hence σ(M)=M and σ is a K-automorphism of M. By Theorem 11.9, M:K is normal.
