Conjecture: Let *G* be a group such that every subgroup of *G* is a normal subgroup. If there exists at least one abelian subgroup of *G*, then there exists a nontrivial homomorphism *f: G ->H* where *H* is some nontreivial subgroup of *G*. No further information about H or the structure of the group. Examples: Z ,Z_n , Quaternion Group hold, simple groups do not hold. If the conjecture is wrong, give a counterexample and suggest what modification is necessary for the conjecture to be correct. ----- Related question: Are there any papers or books that investigate/discuss the relationship between conjugacy classes and normality for the existence of such homomorphisms as follows from the conjecture?