In the paper by Ben Elias et al. (2010), the authors has defined the socle friendly group.
Definition 1: Let $G$ be an arbitrary finite group, and let $\mathcal{T}=\{T \triangleleft G \,|\, T \leq \mathrm{Soc}(G) \} $. We call $G$ socle friendly if for all $H < G, T \in \mathcal{T}$, we have $\mathrm{RC}_G(H \cdot T) = \mathrm{RC}_G(H) \cdot T$, where $\mathrm{RC}_G(H)= \mathrm{Core}_G(H) \cap \mathrm{Soc}(G)$. Here $\mathrm{Core}_G(H)$ is the largest normal subgroup in $G$ containing in $H$.
I understand what is $\mathrm{RC}_G(H)$. Informally saying, it is a subgroup generated by those minimal normal subgroups which are in $\mathrm{Core}_G(H)$
Can someone explain definition 1 and the notation $H \cdot T$ ?