Range-compatible homomorphisms on spaces of symmetric or alternating matrices

In recent works, we have classified the range-compatible group homomorphisms on S when the codimension of S in L(U,V) is small. In the present article, we study the special case when S is a linear subspace of the space Sn(K) of all n by n symmetric matrices: we prove that if the codimension of S in Sn(K) is less than or equal to n−2, then every range-compatible homomorphism on S is local provided that K does not have characteristic 2. With the same assumption on the codimension of S, we also classify the range-compatible homomorphisms on S when K has characteristic 2. Finally, we prove that if S is a linear subspace of the space An(K) of all n by n alternating matrices with entries in K, and the codimension of S is less than or equal to n−3, then every range-compatible homomorphism on S is local.