Let $G$ be a finite group and $H$ a subgroup.

The **normal core** of $H$ in $G$ is $core_G(H) := \bigcap_{g \in G}g^{-1}Hg$

**Definition**: $K$ is a **normal intermediate subgroup** of the inclusion $(H \subset G)$ if $H \subset K \subset G$, and $$\forall g \in G \text{ , } KgH=HgK$$ (This definition is motivated by the prop.3.3 p476 of this paper)

**Examples** : If $H=\{ e \}$ then $K$ is a normal intermediate subgroup iff $K$ is a normal subgroup of $G$.

$H_i$ and $G_i$ are obviously normal intermediate subgroups of the inclusion $(H_i \subset G_i)$, and

$H_1 \times G_2$ and $G_1 \times H_2$ are normal intermediate subgroups of $(H_1 \times H_2 \subset G_1 \times G_2)$.

Let $L$ and $K$ be normal intermediate subgroups of $(H \subset G)$, then $\langle K , L \rangle = KL=LK$.

Question: Is it true that $\forall k \in K$, $k.core_{KL}(K) \cap L \neq \emptyset$ ?

**Remark**: it's true for all the examples above: it's obvious if $H=\{ e \}$ , or if $\{ K , L \} \subset \{ H, G \} $,

and if $G = G_1 \times G_2$, $H = H_1 \times H_2$, $K=H_1 \times G_2$ and $L = G_1 \times H_2$, then $KL=G$ and $\{ e \} \times G_2 \subset core_{KL}(K)$, so if $k \in K$, $k=(h_1,g_2)$ and $(h_1,g_2).(e,g_2^{-1}) = (h_1,e) \in L$.

**Motivation**: This question is (for me) the last step for getting a **Jordan-Hölder** theorem generalized to the inclusions of groups, as explained here in the context of group-subgroup subfactors.

**Definition**: An inclusion $(H \subset G)$ is **simple** if it admits no non-trivial normal intermediate subgroup.

**Examples**: The maximal inclusions are obviously simple, and if $H=\{ e \}$, it's simple iff $G$ is simple.

$(\mathbb{Z}_3 \subset A_5)$ is an example of simple inclusion which is neither maximal nor trivial.