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?

Are there any papers or books that investigate/discuss the relationship between conjugacy classes and normality for the existence of non-trivial homomrphism f:G->H were H is some nontrivial subgroup of G