On the automorphisms of normal subgroups of the collineation group of affine spaces (Q1110090)

Let \(\Sigma\) be a skew field with \(| \Sigma | \geq 3\) and center C. Let V be a right vector space over \(\Sigma\), \(2\leq \dim V<\infty\), H the full collineation group of the affine space over V, T the corresponding translation group, I(H) the group of inner automorphism of H and \(D:=T\cdot (id_ V\cdot \Sigma^*)\) the normal subgroup of dilatations. Generalizing a result of the second author the following are proven. Theorem 1. If \(G\trianglelefteq H\) with \(G\leq D\) and \(T<G\cap (T\cdot id_ VC^*)\), then Aut G\(=I(H)|_ G\) if and only if \(\{k\in \Sigma | id_ Vk\in G\}\) is not contained in a proper sub- skew field of \(\Sigma\). Theorem 2. If \(G\trianglelefteq H\), \(G\nleq D\) and \(T<G\cap D\) then Aut G\(=I(H)|_ G\).