Principal series of a finite super soluble group

Question

Every principal series of a finite super soluble group is a composition series.

I have thought a lot about this and no progress, please help me a little and guide me in right path. Thanks a lot.

Answer 1

Because chief factors are unique (Jordan-Hölder), it is enough for you to prove that a finite supersoluble group $G$ has a chief series with every factor cyclic of prime order.

Now $G$ is finite supersoluble, so there is a series $G = G_0 \supsetneq G_1 \supsetneq \cdots \supsetneq G_n = 1$ where $G_i$ are normal in $G$ and each factor $G_i/G_{i+1}$ is finite cyclic. Use the fact that any subgroup of a cyclic group is characteristic to refine this series into a one where the factors are cyclic of prime order.

Answer 2

A finite group is supersoluble if and only if it has a chief series all of whose factors are cyclic of prime order (see Corollary $5.8$ here). By Jordan-Hölder, every chief series then has cyclic composition factors of prime order, hence every chief series is a composition series.