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Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.

Question: Can any finite lattice be realized as an intermediate subgroups lattice?

Remark: It's true for all the finite distributive lattices (see theorem 2.1 here).

Bjørn Kjos-Hanssen
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Sebastien Palcoux
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2 Answers2

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This is an open problem. See

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Bjørn Kjos-Hanssen
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  • Thank you! I observe that Palfy believes it's not true (see the sentence before theorem 1.7 p 484), so because his paper is more than 20 years old, perhaps we know a counter-example today. – Sebastien Palcoux Feb 08 '15 at 21:41
  • The section 4 of Palfy's paper is dedicated to height 2 lattices. Is it still open in this case? Else, are they some progress since 20 years? – Sebastien Palcoux Feb 08 '15 at 22:11
  • $M_{19}$ is not correct, the list of small values $n$ for which there is not a known representation for $M_{n}$ was, in Palfy's paper (p487): $n=16, 23, 35, 36, 37, 40...$. – Sebastien Palcoux Feb 08 '15 at 23:59
  • I've unaccepted your answer because I've found recent advanced (2013) of the problem (see my answer). Are you agree with this decision? – Sebastien Palcoux Feb 09 '15 at 14:54
  • @SébastienPalcoux Your answer is very interesting! As for whether to accept my answer or your own answer you might go by this: Questions that turn out to be less-famous open problems, but still well-known to experts, are answered by giving a reference to somewhere that the openness of the problem is discussed. -- Noah Snyder, http://meta.mathoverflow.net/a/368/4600 – Bjørn Kjos-Hanssen Feb 09 '15 at 16:14
  • @BjørnKjos-Hanssen For the second bullet point of your answer, I recommend using the link http://www.ams.org/mathscinet-getitem?mr=593011. (The bullet point says "Palfy and Pudlak's result" but your link points a broad overview of the problem... albeit a very nice one.) – William DeMeo May 21 '15 at 20:08
  • @WilliamDeMeo mahalo, I fixed it. – Bjørn Kjos-Hanssen May 21 '15 at 20:48
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The more recent paper I've found about this problem is by Michael Aschbacher in 2013:
Overgroup lattices in finite groups of Lie type containing a parabolic

His introduction is a short survey of the last advanced, he recall Palfy–Pudlak theorem and question, and focus on the following John Shareshian's conjecture. His paper is a first step for a proof of this conjecture:

Let $B_n$ be the subgroups lattice of the cyclic group $C_m$ with $m=p_1p_2 \dots p_n$ square free and $p_i$ prime number.
Let $\mathcal{L}$ be a finite lattice and $\mathcal{L}'$ the poset $\mathcal{L}-\{l,g \}$ with $l$ and $g$ the least and greatest elements of $\mathcal{L}$.

Shareshian's Conjecture: If $\mathcal{L}'$ is a disconnected graph with connected components $(B'_{n_i})_{i=1, \dots , k}$ and $n_i \ge 3$, then $\mathcal{L}$ is not an intermediate subgroups lattice.

The smallest lattice $\mathcal{L}$ coming from this conjecture is the following:

enter image description here

with $k=2$, $n_1=n_2=3$

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Sebastien Palcoux
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    This picture is not quite right. The top and bottom elements from each Boolean algebra should be removed. (That is, the two coatoms and the two atoms in the picture should not be there.) To be clear, I do not know whether the pictured lattice is isomorphic with an interval in the subgroup lattice of a finite group. – John Shareshian Feb 15 '15 at 02:31
  • One more small correction. I believe that the conjecture you attribute to me is a special case of two more general conjectures, one due to Michael Aschbacher and the other due to me. – John Shareshian Feb 15 '15 at 02:39
  • @JohnShareshian: Thank you, I will fix the picture. Michael Aschbacher attributes this conjecture to you in his paper above p72. Do you prefer I call it Aschbacher-Shareshian's conjecture? – Sebastien Palcoux Feb 15 '15 at 12:26
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    Thanks for pointing that out. I won't argue with Michael. – John Shareshian Feb 15 '15 at 16:22