Let G be a group acting on a set X. The action is called faithful if for any g ≠ 1 ∈ G there exists an x ∈ X such that gx ≠ x. That is, only the identity fi xes everything.

Prove the following:

(a) A group G acts faithfully on X if and only if the corresponding homomorphism  Ψ: G -> Aut_Set(X) is injective. Thus, for a faithful action, G is isomorphic to a subgroup of Aut_Set(X).

(b) In general, G/ker Ψ acts faithfully on X. (You will need to de ne how G= ker Ψ acts, and make sure this is an action.)

Let p, q and l be 3 distinct prime numbers, and let G be a finite group. Prove that G is solvable if: