Analyze the abuse of power in Lord of the Flies.

The most obvious perpetrator of power abuse Jack. He openly seeks power over the group from the beginning, leading a group of boys interested in hunting. He wants to overthrow the established order of the group, represented by Ralph. This quest leads him to try to decide whose life is worth saving and who should be sacrificed to the island.
This dilemma begins to emerge after the killing of the first pig: Jack tries to prevent Piggy from eating the meat because he doesn't hunt, even though others who don't hunt are given food. In his power, he quickly decides that Piggy, a physically weaker boy, doesn't deserve to live, and this philosophy pervades his leadership until Piggy's eventual death.
In his leadership, Jack becomes so singularly focused on one idea (hunting) that he becomes blind to the other needs of the group. He also becomes blind to the real dangers of the island and how the boys are slipping into an amoral existence without a code of laws to guide their choices. Under Jack's leadership, the group of boys gathers in a frenzied dance to remind themselves of their goal: "Kill the beast! Cut his throat! Spill his blood!" They are so singularly focused that they kill Simon, mistaking him for the beast. Jack's abuse of power leads the boys to commit murder, and his core group shows no signs of moral regret when the deed is completed.
A more subtle yet still powerful example of an abuse of power is Roger. He is the voice behind the scenes, always pushing others to consider increasingly evil choices. It is Roger who actually kills Piggy, doing so with a "sense of delirious abandonment." When the murder is complete, Samneric regard him with "quiet terror" as he walks by and note that "Roger advanced upon them as one wielding a nameless authority." They convey this sense of Roger's evil leadership earlier in the plot as well when forced to join Jack's tribe:

“You don’t know Roger. He’s a terror.”“And the chief—they’re both—”“—terrors—”“—only Roger—”

Jack may have the position of power, but Roger holds much of the power behind the scenes.
The group puts their collective trust in two boys who aren't capable of rational leadership, and Jack and Roger abuse the power given to them, leading to speedy moral decay and multiple murders.

Not long after the moment the boys are marooned on the island, Jack shows tendencies of wanting to take charge and dictate his will to the boys. Golding hints at his true colors in this way, as Jack is more than ready to establish rules for the sole purpose of meting out punishments. His aggression starts to show itself more fully when he leads the boys on hunting expeditions.
Jack's primary abuse of power lies in his manipulation of the boys' fear of the "beast" and his sheer irresponsibility in attempting to get them rescued. He risks their lives to satisfy his lust for power.
Jack's ultimate abuse of power is manifested in the murders of Simon and Piggy, as well as the attempted murder of Ralph. He has brainwashed the boys to such an extent that he is able to incite them to murder.

A great example of the abuse of power in Lord of the Flies comes in the shape of Jack Merridew. He doesn't see power as something to be used for the common good. On the contrary, he sees it as a great opportunity to control and dominate the other boys. Jack is on a major power trip; he gets off on being in charge. And once he's established himself as virtual dictator of the island, he'll be damned if he lets anyone take power away from him.
One way that Jack abuses power is by playing upon the fear that the other boys have for the mythical beast that's supposed to be roaming the island. Jack knows full well that there's no such creature, but he maintains the lie that it exists to justify his continued control. Jack's clever enough and devious enough to know that if the boys are scared—of the beast, or anything else—they'll be more likely to look to him to be their protector. Jack will then be able to consolidate his dictatorship on the basis of a devil's bargain. Effectively, he says to the other boys, "You give me absolute power and I'll offer you protection." But of course it's Jack and his gang of thuggish choir-boys from which the others need protection.