in $T$ we say the action of $\mathfrak{G}$ on $T$ is [*primitive*]{}, otherwise [*imprimitive.*]{} On the other hand, the family of subsets $\{ \gamma B : \gamma \in \mathfrak{G} \}$ forms a partition of $T$, which is called the [*system of blocks containing $B$*]{} and denoted by $\mathscr{B}$. The action of $\mathfrak{G}$ on the system $\mathscr{B}$ is said to be [*regular*]{} if for any $\gamma \in \mathfrak{G}$, the stabilizer $\mathfrak{G}_B$ fixes $\gamma B$.

Let $B_1,\ldots,B_m$ be a sequence of blocks for $\mathfrak{G}$ such that $B_1 \subsetneq B_2 \subsetneq \cdots \subsetneq B_m \subsetneq B_{m+1} = T$, $B_1$ is a minimal block and $B_i$ is maximal in $B_{i+1}$, [*i.e.,*]{} there is no block $K$ for $\mathfrak{G}$ so that $B_i \subsetneq K \subsetneq B_{i+1}$, $i=1,\ldots,m$. That kind of sequence is said to be a [*block family*]{} of $\mathfrak{G}$. Suppose $\mathscr{B}_i$ is the block system of $\mathfrak{G}$ containing $B_{i}$. We call those systems involved a [*block system family of* ]{} $\mathfrak{G}$, which is denoted by $\mathscr{B}_1 \gneq \mathscr{B}_2 \gneq \cdots \gneq \mathscr{B}_m$. Suppose further that $\mathscr{B}_{i_1},\ldots,\mathscr{B}_{i_r}$ are those systems in the family such that the action of $\mathfrak{G}$ on $\mathscr{B}_{i_j}$ ($j = 1,\ldots,r$) is regular and $\gamma_{1},\gamma_{2},\ldots,\gamma_{r}$ are a group of permutations in $\mathfrak{G}$ 