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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:
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Find x and y in each paarallelogram.
The subsequent force that we want to consider is damping. This force may or may not be there for any specified problem. Dampers work to counteract any movement. There are some w
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4 7 9 6
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three times the first of the three consecutive odd integers is 3 more than twice the third integer. find the third integer.
calculate
what value of k is he sequence 2k+4,3k-7,k+12 are in an arithmetic sequence is
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