Hello

1. G is a group of order $\displaystyle p^n$, with p prime number and n natural number, which acts on a finite set of cardinal X not divisible by p. Show that there is some element x in X such that gx = x for all g in G.

2. If p and q are primes with p <q. Prove that every group G of order pq has only one subgroup of order q normal in G. If q is not congruent to 1 modulo p, show that G is abelian and cyclic.

Thanks, I hope some help