Does every finitely generated group have a maximal normal subgroup? - MathOverflow

Does every finitely generated group have a maximal normal subgroup?

2009-11-04T14:16:16Z

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

Answer by Gabe Cunningham for Does every finitely generated group have a maximal normal subgroup?

2009-11-04T14:39:47Z

Assuming you mean "does a maximal normal subgroup always exist?" (and that you don't care about computing it), here is a way to restate the problem. Notice that if G has no maximal normal subgroups, that means that every proper normal subgroup H of G is contained in a larger proper normal subgroup K of G. In particular, this means that the group G/H must not be finite; if it were, we could only find a finite chain of normal subgroups between H and G. So the question "does a maximal normal subgroup always exist" is the same as "must a finitely generated group have any finite nontrivial quotients?" I'm not sure what the answer to that is, but it seems like a useful restatement.

Answer by Richard Kent for Does every finitely generated group have a maximal normal subgroup?

2009-11-04T14:40:54Z

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 G is < a,b,c,d | a^-1 b a = b^2, b^-1cb = c^2, c^-1dc=d^2, d^-1ad=a^2 >

See Higman, Graham. A finitely generated infinite simple group. 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 ", Journal London Math. Soc, 12 (1937), 120-127.

Answer by Sonia Balagopalan for Does every finitely generated group have a maximal normal subgroup?

2009-11-04T14:49:32Z

Check out the Tarski monster. It is 2-generated and simple.
Unless I misunderstood your question and you exclude infinite simple groups altogether.

Answer by yeshengkui for Does every finitely generated group have a maximal normal subgroup?

2009-11-05T02:32:55Z

So many answers! I'm comletely 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?