# oa.operator algebras – Is there a finite depth irreducible subfactor of prime index and not group-subgroup?

Let $$N subset M$$ be a finite depth unital inclusion of II$$_1$$ factors. By Theorem 3.2 in this paper (Bisch, 1994), if the index $$|M:N|$$ is integer then for any intermediate subfactor $$N subset P subset M$$, $$|M:P|$$ and $$|P:N|$$ are also integer. In particular, if $$|M:N|$$ is prime then there is no proper intermediate (maximal subfactor).

A finite group subfactor remembers the group but a finite group-subgroup subfactor does not always remember the (core-free) inclusion of groups (Kodiyalam-Sunder, 2000). This paper (Izumi, 2002) provides a group-theoretical characterization of isomorphic group-subgroup subfactors. It is easy to see from this characterization that a maximal group-subgroup subfactor remembers the (core-free) inclusion, because the intersection of a core-free maximal subgroup with an abelian normal subgroup is trivial (see here).

A core-free inclusion of finite groups is the same thing than a transitive permutation group, and in the maximal case, replace transitive by primitive. Obviously, at prime degree, a transitive permutation group is always primitive.

So the number of maximal group-subgroup subfactors (up to dual) of index $$n$$ is exactly the number of primitive permutation groups of degree $$n$$, see OEIS/A000019 ($$1, 1, 2, 2, 5, 4, 7, 7, 11, 9, 8, 6, 9, 4, dots$$).

Here is a way to get the corresponding inclusion of finite groups $$H subset G$$ by GAP:

``````gap> NrPrimitiveGroups(5);
5
gap> for i in (1..5) do G:=PrimitiveGroup(5,i);; H:=Stabilizer(G,1);; Print((H,G)); od;
( Group( () ), C(5) )( Group( ( (2,4)(3,5) ) ), D(2*5) )( Group( ( (2,3,4,5) ) ), AGL(1, 5) )
( Group( ( (2,4)(3,5), (3,4,5) ) ), A(5) )( SymmetricGroup( ( 2 .. 5 ) ), S(5) )
``````

The subfactors of index $$2$$ or $$3$$ are given by the groups $$C_2$$, $$C_3$$ and the inclusion $$(S_2 subset S_3)$$, and (dual) principal graphs $$A_4$$, $$D_4$$ and $$A_6$$. At index $$4$$, the subfactors of principal graph $$E_i^{(1)}$$, $$i=6,7,8$$ are all maximal (because $$2$$-supertransitive), but there is only two primitive permutation groups of degree $$4$$, giving the inclusion $$A_3 subset A_4$$ and $$S_3 subset S_4$$, with (dual) principal graphs $$E_i^{(1)}$$, $$i=6,7$$. So the maximal subfactor of index $$4$$ and (dual) principal graph $$E_8^{(1)}$$ is not a group-subgroup subfactor. Finally every finite depth subfactor of index $$5$$ is group-subgroup (Izumi-Morrison-Penneys-Peters-Snyder, 2015).

Conclusion, a finite depth maximal irreducible subfactor of integral index is not always group-subgroup, but the found counter-example (of principal graph $$E_8^{(1)}$$) is not of prime index. So the following question remains:

Question: Is there a finite depth irreducible subfactor of prime index and not group-subgroup?