How does the inspector in The Chrysalids use power/authority to make others suffer injustly and why?
It is the Inspector's job in this district to ensure that any deviations are discovered and mercilessly destroyed according to the regulations laid out in the law. This may mean sacrificing animals or burning crops, or, more disturbingly in the case of humans, sterilising and exiling them into the Fringes and away from acceptable society. Of course, the "justice" that he administers is entirely based on a wrong understanding of the Bible and God's laws, that have somehow been twisted to argue that deviations such as Sophie are dangerous and not actually human. Of course, the biggest victim that we are introduced to in the first few chapters, that foreshadows the kind of treatment that David and others in the group will experience, is Sophie, whose extra toe warrants all the "justice" that the Inspector can bring to bear. Note how he justifies his position to David:
"The Devil sends Deviations among us to weaken us and tempt us away from Purity. Sometimes he is clever enough to make a nearly-perfect imitation, so we have always to be on the look-out for the mistake he has made, however small, and when we see one it must be reported at once."
From the Inspector's point of view, therefore, he is merely upholding the purity of the human species as prescribed in law, but we as readers share David's scepticism of whether this is "justice" or not.