Homomorphic stability of finite groups
The set Hom (G, H) of all homomorphisms from a group G to a group H is a group with respect to the operation of pointwise products iff the images of any two such homomorphisms commute element-wise; in this case, the group is commutative. For finite G and H, we study algebraic properties of this group and of the union Im (G, H) of the images of all homomorphisms from G to H. Let exp(G) be the minimal positive integer n such that xn = 1 for all x £ G, let G' be the commutator subgroup of G, q = exp(G/G'), and let Qq(H) be the subgroup of H generated by all elements of order q. We obtain the following results. If Hom(G,H) is a group, then Qq(H) is commutative and the groups Hom(G,H) and Hom (G/G', Qq(H)) are isomorphic. Conversely, if Qq(H) is commutative and 0(G') = {1} for all ф £ Hom (G, H), then Hom (G, H) is a group. If Im (G, H) is a subgroup of H, then it is endomorphically admissible in H. If G is a finite p-group such that exp(G) = exp(G/G') = q and H is a regular p-group, then Im (G, H) = Qq(H).