Definition: A $T$-group is a group in which normality is a transitive relation.

Definition: A subgroup $H \leq G$ is said to be weakly normal in $G$ if for each $g\in G$, $H^g \leq N_G(H)$ implies that $g\in N_G(H)$.

Definition:: A subgroup $H$ is weakly pronormal in $G$, if for any subgroups $K$ and $L$ of $G$ such that $H \leq K \unlhd L \leq G$, we have $L = N_L(H)K$

The following is a characterisation of a finite solvable $T$-group presented in journal paper by P.V Gavron

Let $G$ be a finite group. Then the following are equivalent:

(a) $G$ is a solvable $T$-group;

(b) All subgroups of $G$ are weakly normal in $G$;

(c) All subgroups of $G$ are weakly pronormal in $G$.

I have managed to show that (a) implies (b) and (c) implies (a). I am having a problem showing that (b) implies (c).

Asumme that $H \leq K \unlhd L \leq G$. We will show that $H$ is weakly pronormal in $G$. It suffices to show that $L \leq N_L(H)K$.


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