What is the purpose of the Jack's, Ralph's, and Simon's expedition?
I assume you are referring to the expedition taken by Jack, Ralph, and Simon in the first chapter of Lord of the Flies since that is when the three of them go together. At this point in the story, the split between the boys is not noticeable other than the fact that Jack is in charge of the choir because he was chapter chorister and head boy at school. The purpose of this journey is to examine the island that they've all just crash landed on. They want to see, for certain, that it is an island and what it holds for them. Even as boys they realize that they will need fresh water and food. These three boys decide to go because Ralph is the newly elected chief, Jack has put himself in charge of the hunter group, and Simon is among the older of the boys on the island. Symbolically, Simon is the mystic, or spiritual person, among the boys, and it is fitting that the protagonist (Ralph), the antagonist (Jack), and the boy who is the first to recognize the true source of evil on the island should be the ones to initially explore the island. On this expedition, the three boys do, indeed, realize they are on an island and that if they hope to get rescued, they must make a signal of some sort. This expedition also serves to allow for the situation created when the three boys encounter a pig. Jack attempts to stab it, but he is still too civilized to go through with it. This highlights the change we see later in Jack, from a civilized boy to a savage.