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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 fixes 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 dene 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:
Recall also which value of the derivative at a specific value of t provides the slope of the tangent line to the graph of the function at that time, t. Thus, if for some time t the
2+2=
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circumference of a circle
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For the layman, a "function" indicates a relationship among objects. A function provides a model to describe a system. Economists refer to deman
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