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The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:

Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $G$ a simple group?

Remark: For $\vert G \vert \leq 4000000$, we have found (by GAP):

• $A_8$ (of order $20160$) with a subgroup of index $315$,
• $PSU(3,5)$ (of order $126000$) with a subgroup of index $6000$,
• $PSp(6,2)$ (of order $1451520$) with a subgroup of index $2835$,
• $PSU(4,3)$ (of order $3265920$) with a subgroup of index $25515$.

Can we have a classification in general?

There is a large class of examples given by the BN-pairs, as pointed out in Example 4.21 of this paper:

Let $G$ be a finite group with a BN-pair, $H$ be the corresponding Borel subgroup and $(W, S)$ be the associated Coxeter system. Let $n :=|S|$ be the rank of the BN-pair. Then the interval $[H, G]$ is Boolean of rank $n$. Any finite simple group $G$ of Lie type (over a finite field of characteristic $p$) admits a BN-pair (except Tits group). If moreover, $G$ is a Chevalley group, then $n$ is the number of vertices in its Dynkin diagram.

The above interval with $G = A_8 $ or $ PSp(6,2)$ comes from a BN-pair (where $G$ is the Chevalley group $A_3(2)$ or $C_3(2)$), whereas the one with $G = PSU(3,5) $ or $ PSU(4,3)$ does not.

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:

Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $G$ a simple group?

Remark: For $\vert G \vert \leq 4000000$, we have found (by GAP):

• $A_8$ (of order $20160$) with a subgroup of index $315$,
• $PSU(3,5)$ (of order $126000$) with a subgroup of index $6000$,
• $PSp(6,2)$ (of order $1451520$) with a subgroup of index $2835$,
• $PSU(4,3)$ (of order $3265920$) with a subgroup of index $25515$.

Can we have a classification in general?

There is a large class of examples given by the BN-pairs, as pointed out in Example 4.21 of this paper:

Let $G$ be a finite group with a BN-pair, $H$ be the corresponding Borel subgroup and $(W, S)$ be the associated Coxeter system. Let $n :=|S|$ be the rank of the BN-pair. Then the interval $[H, G]$ is Boolean of rank $n$. Any finite simple group $G$ of Lie type (over a finite field of characteristic $p$) admits a BN-pair (except Tits group). If moreover, $G$ is a Chevalley group, then $n$ is the number of vertices in its Dynkin diagram.

The above interval with $G = A_8 $ or $ PSp(6,2)$ comes from a BN-pair (where $G$ is the Chevalley group $A_3(2)$ or $C_3(2)$), whereas the one with $G = PSU(3,5) $ or $ PSU(4,3)$ does not.

Edit (29/08/2021): See the recent paper Boolean lattices in finite alternating and symmetric groups by Andrea Lucchini, Mariapia Moscatiello, Pablo Spiga and myself.

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:

Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $G$ a simple group?

Remark: For $\vert G \vert \leq 4000000$, we have found (by GAP):

• $A_8$ (of order $20160$) with a subgroup of index $315$,
• $PSU(3,5)$ (of order $126000$) with a subgroup of index $6000$,
• $PSp(6,2)$ (of order $1451520$) with a subgroup of index $2835$,
• $PSU(4,3)$ (of order $3265920$) with a subgroup of index $25515$.

Can we have a classification in general?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:

Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $G$ a simple group?

Remark: For $\vert G \vert \leq 4000000$, we have found (by GAP):

• $A_8$ (of order $20160$) with a subgroup of index $315$,
• $PSU(3,5)$ (of order $126000$) with a subgroup of index $6000$,
• $PSp(6,2)$ (of order $1451520$) with a subgroup of index $2835$,
• $PSU(4,3)$ (of order $3265920$) with a subgroup of index $25515$.

Can we have a classification in general?

There is a large class of examples given by the BN-pairs, as pointed out in Example 4.21 of this paper:

Let $G$ be a finite group with a BN-pair, $H$ be the corresponding Borel subgroup and $(W, S)$ be the associated Coxeter system. Let $n :=|S|$ be the rank of the BN-pair. Then the interval $[H, G]$ is Boolean of rank $n$. Any finite simple group $G$ of Lie type (over a finite field of characteristic $p$) admits a BN-pair (except Tits group). If moreover, $G$ is a Chevalley group, then $n$ is the number of vertices in its Dynkin diagram.

The above interval with $G = A_8 $ or $ PSp(6,2)$ comes from a BN-pair (where $G$ is the Chevalley group $A_3(2)$ or $C_3(2)$), whereas the one with $G = PSU(3,5) $ or $ PSU(4,3)$ does not.

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:

Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $G$ a simple group?

Remark: For $\vert G \vert \leq 4000000$, we have found (by GAP):

• $A_8$ (of order $20160$) with a subgroup of index $315$,
• $PSU(3,5)$ (of order $126000$) with a subgroup of index $6000$,
• $PSp(6,2)$ (of order $1451520$) with a subgroup of index $512$,
• $PSU(4,3)$ (of order $3265920$) with a subgroup of index $25515$.

Can we have a classification in general?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:

Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $G$ a simple group?

Remark: For $\vert G \vert \leq 4000000$, we have found (by GAP):

• $A_8$ (of order $20160$) with a subgroup of index $315$,
• $PSU(3,5)$ (of order $126000$) with a subgroup of index $6000$,
• $PSp(6,2)$ (of order $1451520$) with a subgroup of index $2835$,
• $PSU(4,3)$ (of order $3265920$) with a subgroup of index $25515$.

Can we have a classification in general?