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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 -> AutSet(X) is injective. Thus, for a faithful action, G is isomorphic to a subgroup of AutSet(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:
the first question should be done using the website given (www.desmos.com/calculator )and another good example to explain using the graph ( https://www.desmos.com/calculator/ydimzr
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Thorwarth M., Arisha, A. and Harper P., (2009) Simulation model to investigate flexible workload management for healthcare and servicescape environment, Proceedings of the 2009 Win
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