Given an infinite group which is finitely generated, is there a proper maximal normal subgroup?



Does every finitely generated group have a maximal normal subgroup?
If you mean nontrivial maximal normal subgroup (not 1 or the whole group), then the answer is no.


Higman constructed a finitely generated infinite group $G$ with no subgroups of finite index. You then get a finitely generated group with no nontrivial normal subgroups by taking the quotient by a maximal normal subgroup.


[Higman's group](https://en.wikipedia.org/wiki/Higman_group) $G$ is $\langle a,b,c,d | a^{-1} b a = b^2, b^{-1}cb = c^2, c^{-1}dc=d^2, d^{-1}ad=a^2 \rangle$


See Higman, Graham. [A finitely generated infinite simple group](https://dx.doi.org/10.1112/jlms/s1-26.1.61). J. London Math. Soc. 26, (1951). 61--64. 


Edit:


If you mean does it have a proper maximal normal subgroup, then the answer is yes:


Finitely generated groups have a (possibly trivial) maximal normal subgroup. Higman's reference for this is B.H. Neumann, "[Some remarks on infinite groups](http://jlms.oxfordjournals.org/cgi/reprint/s1-12/2/120) ", Journal London Math. Soc, 12 (1937), 120-127.


