# Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there.

Recall that a subfactor is Dedekind if all its intermediate subfactors are normal.

A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice of intermediate subfactors is a single chain: $N=P_0 \subset P_1 \subset \dots \subset P_{r-1} \subset P_r= M$, with $(P_i \subset P_{i+1}) \simeq (P_0 \subset P_1)$.

Question: Are the HSC subfactors Dedekind ?

Remark: The maximal subfactors are obviously HSC and Dedekind.

Group-subgroup case
An inclusion of groups $(H \subset G)$ is HSC if its lattice of intermediate subgroups is a single chain $H=K_0 \subset K_1 \subset \dots \subset K_{r-1} \subset K_r = G$, with $(K_i \subset K_{i+1}) \sim (K_0 \subset K_1)$.

Remark: If $r \ge 2$ then $G$ is not a finite simple group.
Proof: Recall that $(A \subset B) \sim (C \subset D)$ iff $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$.
Now if $r \ge 2$ then $(K_{r-2} \subset K_{r-1}) \sim (K_{r-1} \subset G)$, so if $G$ is a finite simple group,
then $\vert G \vert \le \vert K_{r-1} \vert$, contradiction.

Recall that $K$ is a normal intermediate subgroup of $(H \subset G)$ if:
$(H \subset K \subset G)$ and $\forall g \in G$, $HgK=KgH$.
So in this case, the question is:

If $(H \subset G)$ is HSC and $H<K<G$, then, is it true that $HgK=KgH$, $\forall g \in G$ ?

Experiment for $r=2$: It's checked for $\vert G \vert \le 2000$ (except $512$, $1024$, $1536$ and $1792$).
It's also true if $[G:H] \le 30$ (see this comment of D. Holt).

Remark: This experimental answer is very hopeful for a positive answer in the group-subgroup case, and a generic proof would be very helpful for finding a proof for the general subfactor case.

Natural subfactors theory (optional part)

The cyclic subfactors (i.e. distributive lattice) is a too large class for being an good quantum arithmetic (i.e. generalizing the natural numbers), because several of the following natural properties of the cyclic group subfactors $(R^{\mathbb{Z}_n} \subset R)$, $n \in \mathbb{N}$ , are not all checked by the cyclic subfactors (see for ex. here):

(1) Irreducible: $N' \cap M = \mathbb{C}$
(2) Cyclic: the lattice $\mathcal{L}(N \subset M)$ is distributive
(3) Dedekind: all the intermediate $P$ are normal
(4) Jordan-Hölder: $(P_1 \cap P_2 \subset P_1) \simeq (P_2 \subset P_1P_2)$, for all $P_i$ normal.
(5) It decomposes (for $\otimes$) into $HSC$ subfactors.

Definition: A natural subfactor is an irreducible cyclic subfactors generated (for $\otimes$) by the $HSC$ irreducible subfactors (note that tensor products of finitely many cyclic subfactors are cyclic iff there is no depth $2$ maximal intermediate in common (see here)).

Remark: The maximal and $HSC$ irreducible subfactors are obviously natural subfactors. The natural subfactors check (1), (2), (4), (5) and up to a positive answer to the main question, they also check (3).
If there are non-Dedekink $HSC$ subfactors, we restrict the definition of natural subfactors to those generated by Dedekink $HSC$. So the natural subfactors theory is a good quantum arithmetic.

Group-subgroup case: A transitive permutation group $G$ is $HSC$ if $(G_1 \subset G)$ is a $HSC$ inclusions of groups. It's natural if $(G_1 \subset G)$ decomposes into a direct product of finitely many $HSC$ inclusions of groups and if $\mathcal{L}(G_1 \subset G)$ is distributive. The primitive and $HSC$ groups are obviously natural.

Table: the number of transitive groups of degree $n\le 30$, for several classes.

$\begin{array}{c|c} n &2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18 \newline \hline T &1&2&5&5&16&7&50&34&45&8&301&9&63&104&1954&10&983 \newline \hline D &1&2&4&5&15&7&39&32&44&8&249&9&62&104&1055&10&894 \newline \hline N &1&2&4&5&6&7&24&26&14&8&18&9&11&16&681&10&38 \newline \hline H &1&2&4&5&4&7&22&24&9&8&6&9&4&6&667&10&4 \newline \hline P &1&2&2&5&4&7&7&11&9&8&6&9&4&6&22&10&4 \end{array}$
$\begin{array}{c|c} n & 19&20&21&22&23&24&25&26&27&28&29&30& Total \newline \hline T &8&1117&164&59&7&25000&211&96&2392&1854&8&5712&40225 \newline \hline D &8&923&163&58&7&15627&208&95&2151&1541&8&5461&28798 \newline \hline N &8&33&23&12&7&75&102&16&689&46&8&58&1957 \newline \hline H &8&4&9&4&7&5&88&7&643&14&8&4&1593 \newline \hline P &8&4&9&4&7&5&28&7&15&14&8&4&230 \end{array}$

T: Transitive permutation group $G$
D: Distributive lattice of $(G_1 \subset G)$
N: Natural transitive permutation group
H: Homogeneous single chain inclusion $(G_1 \subset G)$
P: Primitive permutation group (i.e. maximal inclusion $(G_1 \subset G)$)

There are $40225$ transitive groups $G$ of degree $\le 30$, whose $28798$ with $\mathcal{L}(G_1 \subset G)$ distributive. Conclusion (up to $\sim$) more than $70$% of the inclusions of groups of index $\le 30$ are distributive lattice, so we can interpolate that in the general subfactors theory, the cyclic subfactors are majority !

The above approach of the natural subfactors is "bottom-up".

Question: Is there a top-down definition (i.e. without using the $HSC$ subfactors)?

Remark: The subfactor $(R^{\mathbb{Z}_{15} \rtimes \mathbb{Z}_2} \subset R^{\mathbb{Z}_2})$ is irreducible, cyclic, Dedekind and Jordan-Hölder but not natural, because it is prime for $\otimes$ but not $HSC$.