# What groups have a second maximal subgroup below exactly four maximal subgroups?

I am looking for a finite group $G$ with the following property: there is a (core-free) subgroup $H < G$ such that the interval $\{ K : H < K < G\}$ in Sub[$G$] contains exactly four maximal subgroups. (In other words, $[H, G] \cong M_4$.)

I have used GAP to search for such groups and, to my surprise, I could find only three: $S_3$, $C_3 \times C_3$, and $(C_3 \times C_3) : C_2$. So far, all other examples reduce to these after modding out by a normal subgroup (so they are not examples if we require $H$ be core-free).

I've searched through most of the groups of order less than 960. Though, I can't promise my GAP code is free of bugs that may be causing me to miss something.

Question: Does anyone know of other finite groups, besides $S_3$, $C_3 \times C_3$, and $(C_3 \times C_3) : C_2$, with an upper interval isomorphic to $M_4$? (If not, I would welcome any ideas that could help explain why this should be a rare phenomenon.)

• Given your examples, it looks like the orders of the maximal subgroups above H are at most 3 times the order of H. I suggest looking at how the maximal subgroups acting on left cosets of H must look, and show that there are only finitely many choices for these. You may end up with something that will work for M_n type upper intervals. Gerhard "Ask Me About System Design" Paseman, 2011.05.01 May 2, 2011 at 6:46
• Did you take a look Roland Schmidt's book "Subgroup Lattices of Groups"? (I don't have a copy of it available here, but this would be my first reference.) May 2, 2011 at 13:58

(Updated to reflect John Shareshian's excellent answer and suggestions.)

Yes, there are infinitely many examples.

First off, some negative results: If $[G:H] ≤ 31$, then $G/Core(G,H)$ acting on H is one of your three examples. If H is contained in a core-free maximal subgroup M with $[G:M] ≤ 50$ (subject to a few caveats), then in fact G is your first example. These "$M$" correspond to John Shareshian's type (2) examples.

However, I think your search of small groups must have had an error:
The group $G =$ SmallGroup(648,725) has Sylow $2$-subgroup $H$ that is core-free, and the interval $G/H$ is the bounded version of a $4$-element anti-chain.
This corresponds to the next smallest $H$ (after $H≅2$) in John Shareshian's type (1) examples.

As a general comment, no interval is so rare that it only occurs finitely many times:

Given any interval $[G/H]$ there is an anti-isomorphic interval in a wreath product of the permutation group $(G,H)$ with a non-abelian simple group.

Applying the construction an even number of times produces an infinite sequence of examples with a given (core-free) interval. In particular, there is an example with $|G|=279936000000$ and $|H|=360$, with $H$ core-free and the interval $G/H$ the bounded version of a $4$-element anti-chain, $M_4$. This takes the "seed" $(G,H)$ to be the regular representation of the non-abelian group of order $6$, and the non-abelian simple group of order $60$.

Here's what I've found in the literature (both of which are referenced in Roland Schmidt's book; I did not find much else):

Kurzweil (1985) puts your second and third example into context ($G/N=H$ acting on an isotypic semisimple module $N$, so that the interval $[G/H]$ is a projective space). Your first example is just "small", I think. More importantly, it gives the method of replicating examples using wreath products as the second example on page $148$.

Heineken (1987) pins down the structure of solvable $G$ with second maximal $H$ such that the interval $[G/H]$ is not a (bounded) antichain, $M_n$. He has some results that say $n−1$ is usually a prime power.

John Shareshian points out that Baddeley–Lucchini (1997) is dedicated to the question of which $M_n$ can occur as intervals in the subgroup lattice of a finite group. In particular, this shows that the $n−1$ being a prime power result is definitely restricted to solvable groups: Feit showed both $n=7$ and $n=11$ occur. The paper is definitely focused on the non-solvable case.
The Math Review is also really good.

Kurzweil, Hans. "Endliche Gruppen mit vielen Untergruppen." J. Reine Angew. Math. 356 (1985), 140–160. MR779379 DOI:10.1515/crll.1985.356.140 GDZ:GDZPPN002202190

Heineken, H. "A remark on subgroup lattices of finite soluble groups." Rendiconti del Seminario Matematico della Università di Padova, 77 (1987), 135-147 MR904616 NUMDAM:RSMUP_1987__77__135_0

Baddeley, Robert; Lucchini, Andrea "On representing finite lattices as intervals in subgroup lattices of finite groups." J. Algebra 196 (1997), no. 1, 1–100. MR1474164 DOI:10.1006/jabr.1997.7069

• Thank you for your helpful explanations. I have a few questions about your answer before I accept it. 1. How do you know that we must get one of these three examples when [G:H] $\leq$ 31? Moreover, how do you get that it must be the first example when [G:M] $\leq$ 50? If you think these are easy observations, just say so, and I will think about it some more. 2. In your wreath product construction, you say "permutation group (G,H)" -- is this short-hand for "G acting on cosets of H (embedded in Sym([G:H]))"? (to be continued) May 2, 2011 at 22:39
• 3. You say, "given any interval [G/H] there is an isomorphic interval in a wreath product..." Do you mean there is a DUALLY isomorphic interval? This is a result of Kurzweil with which I'm familiar, but maybe you're thinking of a different construction. (Of course, for this self-dual example, either would be ok.) 4. I'm familiar with a result of Heineken from the paper you mention (though I learned it 2nd hand). It says, if G is solvable and $[G/H] \cong M_n$ (n>2), then n-1 is a prime-power. This result doesn't help here, since n=4. (Still, it's a useful/relevant reference -- thanks!) May 2, 2011 at 22:52
• 5. Finally, I think I see how you might have constructed your example with |G|=279936000000 and |H|=360 but could you please say "yes" ("no") if the following is right (wrong)? Start with one of the three original examples, e.g. $S=(C_3\times C_3)\ltimes C_2$, embedded in Sym(9) (action on cosets). Take any nonabelian simple group $A$ and form the wreath product $A\wr S = A^9 \ltimes S$. If $\Gamma$ is the "diagonal" subgroup of $A^9$, then our original interval should be (the dual of) $[(A\wr S) / (\Gamma \wr S)]$, right? May 2, 2011 at 23:36
• 1. I just asked GAP to check. For [G:H] ≤ 31, this means G/Core(G,H) is in the transitive groups library. For [G:M] ≤ 2499, this means G/Core(G,M) is in the primitive groups library, but subgroup lattice computations by brute force are hard if G is too big. I had to intervene for a few of the calculations, and the patterns suggested to me I was looking at the wrong type of groups (for instance, neither your second nor third example are of this type). 2. Yes, exactly. May 3, 2011 at 4:01
• 3. The construction I mention is the bottom of page 148 of Kurzweil (1985). If it is dual, then just apply it twice to still get infinitely many examples for any interval. Mn is self-dual, so there is no problem. 4. No problem. I meant this more of as a complement to GP's comment that you could handle general Mn, since Heineken had studied some general Mn. 5. Yes, almost exactly. I think I used your first example in Sym(6) to keep the order down. 60^6 * 6 = 279936000000 versus 60^9 * 9 or 60^9 * 18. I used Magma to sanity check the calculation. May 3, 2011 at 4:05

I think that with some work you can characterize the examples where $G$ is solvable as follows. (I did not check carefully, so you can take this with a grain of salt.)

If $H$ has trivial core in $G$ and $[H,G]$ is $M_4$, then one of the following holds:

1) $G$ is the semidirect product $H(V+V)$, where $V$ is an irreducible $F_3[H]$-module such that the only elements of $GL(V)$ commuting with $H$ are the scalar transformations $1$ and $-1$.

(This case occurs when every maximal subgroup containing H has nontrivial core in $G$. Your example $C_2:C_3xC_3$ is of this type. This condition should be sufficient and you can construct tons of examples of this type without too much trouble.)

2) $G$ is the semidirect product $MV$, where $V$ is an irreducible $F_3[M]$-module. There is a $1$-dimensional subspace $W$ of $V$ such that $H=N_M(W)=C_M(W)$, and $C_V(H)=W$.
In particular, no element of $M$ has $W$ as an eigenspace with eigenvalue $-1$.

(This case occurs when some maximal $M$ containing H has trivial core in $G$. Your example $S_3$ is of this type. I don't know if there are lots of examples with $H$ maximal in $M$)

I can recommend that if you are interested in understanding how to find intervals of type M_n, you look at the work of Andrea Lucchini, including his joint paper with Robert Baddeley in Journal of Algebra. There is also recent work of Michael Aschbacher on the interval representation problem in general that is most certainly worth examining.

• In part 1, you just mean V is absolutely irreducible, right? Must V be faithful as well? May 5, 2011 at 16:08
• I'd be interested in a few examples of type 2. They seemed rare, but perhaps I stopped looking too early. May 5, 2011 at 16:10
• Thank you very much for these interesting suggestions. I hadn't been thinking about semidirect products this way, but it looks like I should be. As for Baddeley, Borner, Lucchini, yes, I'm familiar with their work, as well as your recent work with Aschbacher. However, I'm particularly interested in M_4 because, afaik, no one has found a "hereditary" M_4. It seems that, if such an M_4 exists, we can assume it's an interval in a subgroup lattice. If we could then characterize such groups, perhaps we could prove there's no hereditary M_4... a longshot, but your suggestions are a great start! May 6, 2011 at 9:37
• 1) Jack, right, V is absolutely irreducible. It seems if V is not faithful then H does not have trivial core in G. Also, in case 2, the condition that no element of M have eigenvalue -1 on W need not hold if W=V, as in G=S_3. 2) William, I am trying to understand the definition of hereditary in the paper of Hegedus-Palfy. Is there a better place? Also, I would recommend Aschbacher's solo papers over our joint one. May 6, 2011 at 15:49
• It seems very likely that to characterize all [H,G]=M_4, one would need to use the CFSG. For a start, in his thesis Alberto Basile showed that one cannot have [H,G]=M_4 if G=S_n or A_n, n \geq 5. (I think - see Theorem D. If I understand what is meant by "examples of Feit and Palfy", then we never see M_4.) The thesis is on the arxiv. May 6, 2011 at 15:56