Does every finitely generated group have a maximal normal subgroup?

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

• Two trivial comments, not worth a full answer. (1) Obviously, the group itself is a maximal normal subgroup. I assume you want to exclude that. (2) The language "can one find" strikes me as a little ambiguous: I don't know whether you mean "does one exist?" or "is one computable/constructible?" I'm sure the answer to the second is "no", because almost everything in finitely presented groups is not computable. The first question is a nice problem. – David E Speyer Nov 4 '09 at 14:29
• I think that the title of the question (if indeed not the body) suggests that the question is "does one exist". – José Figueroa-O'Farrill Nov 4 '09 at 14:35
• Yes, Iam interested in lesser goal : does one exists. – arun s Nov 4 '09 at 14:38
• Regarding David Speyer's comment above, I think it's common practice to use "maximal" as shorthand "maximal proper" in many contexts. (For example, this is how Hungerford's Algebra uses it in "maximal ideal", etc.) – Mark Meckes Nov 4 '09 at 14:40
• I assume the thing you're worried about is the union of a chain of proper normal subgroups being whole group. If you know that doesn't happen, you get existence of a maximal one by Zorn's lemma. – Anton Geraschenko Nov 4 '09 at 14:51

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 $\langle a,b,c,d | a^{-1} b a = b^2, b^{-1}cb = c^2, c^{-1}dc=d^2, d^{-1}ad=a^2 \rangle$

See Higman, Graham. A finitely generated infinite simple group. 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.

• But I think {1} should count as nontrivial. Otherwise, any finite simple group is a counter-example. I think an interesting question would be "Does Higman's group have a maximal proper normal subgroup?" – David E Speyer Nov 4 '09 at 14:47
• Note that this uses the fact that 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. – Autumn Kent Nov 4 '09 at 14:48
• Our comments crossed. So I change my answer to yes (see Neumann's paper). – Autumn Kent Nov 4 '09 at 14:49
• @arun s: Why "thank you anyway..."? Richard's first comment answers your question affirmatively. @Richard: you should edit that comment into the answer, since it's the "real" answer – Anton Geraschenko Nov 4 '09 at 14:58
• Nice! And Neumann's paper is online jlms.oxfordjournals.org/cgi/reprint/s1-12/2/120 . – David E Speyer Nov 4 '09 at 15:16

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

• Downvoted: If the "proper" in the initial question does include the trivial subgroup, then this example does not qualify. If it does not, than the question is not interesting as any finitely-generated simple group will do, trivially. In any case, nothing from the monster properties is required here. – Pasha Zusmanovich Nov 5 '09 at 18:43
• You are right, of course. I had been thinking about this group sometime before I saw the question, hence the unnecessarily complicated "counter"example! – Sonia Balagopalan Nov 5 '09 at 20:58

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?

• A maximal subgroup M of G is either normal or self-normalizing. In the latter case, the union U of a chain of normal subgroups contained in M will be normal and cannot equal G, as it lies in M. So U will be a maximal normal subgroup of G. – Autumn Kent Nov 5 '09 at 3:19
• This argument is flawed. For example, $Sym(6)$ is a maximal subgroup of $Sym(7)$. The only normal subgroup of $Sym(7)$ which is contained in $Sym(6)$ is the identity subgroup, which clearly is not a maximal proper normal subgroup. – Simon Thomas May 17 '10 at 0:31
• Oops, you're right, Simon. – Autumn Kent May 17 '10 at 2:18
• Neumann states the result for maximal subgroups, but his proof works equally well for maximal normal subgroups. – Greg Kuperberg Jun 17 '17 at 1:16

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.

• Well, it's not equivalent -- a group with an infinite simple group as a proper quotient has a maximal normal subgroup. – Tom Church Nov 4 '09 at 16:57
• Yes, good point. What I meant to say is that if finitely generated groups always have at least one finite nontrivial quotient, then they must have a proper maximal subgroup. – Gabe Cunningham Nov 4 '09 at 18:54