I am struggling to fully understand and tackle the following exercise:
If $G$ is a finite group acting transitively on a finite set $\Omega$, then the permutation representation $k\Omega$, with $\mathrm{char}k = p$, is a semisimple $kG$-module if and only if $p\nmid |\Omega|$.
Now, from Wikipedia I gather that the permutation representation is the pair $(k\Omega,\rho)$ with $\rho:G\rightarrow \mathrm{Sym}(\Omega)$, and extended to be $k$-linear.
I see that the transitivity of the action of $G$ implies that the action is fully determined by its effect on any one element of $\Omega$ - does this imply that $G$ is cyclic?
For the 'only if' direction, I believe I need to assume $p\mid |\Omega|$, and construct a submodule without complement, to contradict complete reducibility. Not sure how, though.
I have no idea how to deal with the 'if' direction.
I feel I am missing some important idea on how to use all this finiteness.
Note: this is at the beginning of a course in Representation Theory, hence there is not much machinery - I can't see how to use the Jordan-Hölder Theorem, the Schur's lemma does not apply; maybe I need to look at the central characters of $k\Omega$..?