Q:

Why there exists an element $ y\in G\setminus‎ L $ such that $ K/L_{G} = (\langle y \rangle L_{G})/L_{G} $?

Let $ G $ is finite supersoluble group. Then $ G $ has a chief series. Let $ L $ be a subgroup of $ G $ and $ K/L_{G} $ a chief factor of $ G $ where $ L_{G} $ is the core of $ L $ in $ G $. Since $ G $ is supersoluble, $ K/L_{G} $ is a cyclic group with prime order. Clearly, $ K $ is not contained in $ L $. Why there exists an element $ y\in G\setminus‎ L $ such that $ K/L_{G} = (\langle y \rangle L_{G})/L_{G} $?

A:

Since $G$ is supersoluble, its chief factor is $1$ dimensional. So $|G|=|L_G||K/L_G|$. Since $|K/L_G|=p$, $K/L_G$ is generated by $yL_G$, $y\in G, y\notin L$, for otherwise $K/L_G=L_G$.

