1. G is a group of order , 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.
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