TITLE: Does every finitely generated group have a maximal normal subgroup?
QUESTION [12 upvotes]: Given an infinite group which is finitely generated, is there a proper maximal normal subgroup?

REPLY [2 votes]: So many answers! I'm completely lost. The paper of "B.H. Neumann, "Some remarks on infinite groups", Journal London Math. Soc, 12 (1937), 120-127" stated results for the existence of maximal subgroups, not maximal normal subgroup. Is this existence question of nontrivial normal subgroup still unsolved?