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.

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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?