Suppose $G \leq S_n$ and $S$ is a strong generating set of G. Before starting, I'll describe how $S$ is constructed in my particular case.
- Suppose we have a subgroup chain $G = G_0 \geq G_1 \geq G_2 \geq \dotso \geq G_m = \{e\}$. Consider $G_i / G_{i+1}, 0 \leq i \leq m-1$. For each $i$ pick $e$ from $G_i$ and one element from each of the other cosets in $G_i / G_{i+1}$ forming a table of elements. This table of elements represents a strong generating set for G.
- Let $G_{1...i} \leq G$ be the subgroup consisting of all those elements $g \in G$ such that $g(1)=1,\dotso g(i)=i$. Then $G \geq G_1 \geq G_{12} \geq \dotso G_{1...n} = \{e\}$ is a subgroup chain. $S$ is constructed based on this subgroup chain.
My problem is to construct a method that allows to find a generating set of a proper subgroup of $G$ by modifying $S$. The limitation is that this method should allow me to select a random subgroup for G. The goal is to repeat it recursively and construct a random subgroup chain (meaning that the individual subgroups are picked randomly based on some rule). I know that if $\langle S \rangle = G$ then $\forall \emptyset \neq S' \subset S, \langle S' \rangle \leq G$. So the first idea is to remove some elements from $S$ and see what happens. The problem is that there's no guarantee that this subset will generate a proper subgroup of $G$.
I'm not really sure how to proceed. I don't know whether there is a general approach or one suitable for the specific strong generating set I have. The problem shouldn't be hard, but I need some direction. Any help would be appreciated. Thanks in advance!