There is an equivalence relation between inclusion of finite groups coming from the world of subfactors:

**Definition**: $(H_{1} \subset G_{1}) \sim(H_{2} \subset G_{2})$ if $(R^{G_{1}} \subset R^{H_{1}})\cong(R^{G_{2}} \subset R^{H_{2}})$ as subfactors.

Here, $R$ is the hyperfinite $II_1$ factor (a particular von Neumann algebra), and the groups $G_1$ and $G_2$ act by outer automorphisms. The notation $R^G$ refers to the fixed-point algebra.

**Theorem**: Let $(H \subset G)$ be a subgroup and let $K$ be a normal subgroup of $G$, contained in $H$, then:

$(H \subset G) \sim (H/K \subset G/K)$. In particular, if $H$ is itself normal: $(H \subset G) \sim (\{1\} \subset G/K) $

**Theorem** : $(\{1\} \subset G_{1}) \sim(\{1\} \subset G_{2})$ iff $G_1 \simeq G_2$ as groups.

**Remark** : the relation $\sim$ remembers the groups, but not necessarily the subgroups:

**Exemple** (Kodiyalam-Sunder p47) : $(\langle (1234) \rangle \subset S_4) \sim (\langle (13),(24) \rangle \subset S_4)$

Is there a purely group-theoretic reformulation of the relation $\sim$ ?

**Motivations**: See here and here.

**Some definitions:** A *subfactor* is an inclusion of factors. A *factor* is a von Neumann algebra with a trivial center. The *center* is the intersection with the commutant. A *von Neumann algebra* is an algebra of bounded operators on an Hilbert space, closed by taking bicommutant and dual. Here, $R$ is the hyperfinite $II_{1}$ factor. $R^{G}$ is the subfactor of $R$ containing all the elements of $R$ invariant under the natural action of the finite group $G$. In its thesis, Vaughan Jones shows that, for all finite group $G$, this action exists and is unique (up to outer conjugacy, see here p8), and the subfactor $R^{G} \subset R$ completely characterizes the group $G$. See the book *Introduction to subfactors* (1997) by Jones-Sunder.