I am trying to find the composition series of the group $\mathbb F_{p^k}$, where $p$ is prime , and $k\ge 1$. From Jordan Hölder equation it has length $k$ , i am quite confused , It looks quite natural to write $${e}\subset \mathbb F_p \subset...........\mathbb F_{p^k}$$
And each of them are definitely the maximal subgroups ( my doubt is how do i follow normality of every subgroup and the abelian nature of factor groups ie $\mathbb F_{p^i}/\mathbb F_{p^{i-1}}$ . Thank you .