The Action of a Permutation Group

Abstract

The next three chapters are primarily devoted to studying primitive groups. Primitive groups play an important role as building blocks, particularly in the study of finite permutation groups. Frequently, we can carry out a series of reductions: from the general case to the transitive case by examining the action of the group on its orbits and its point stabilizers, and then from the transitive imprimitive case to the primitive case by studying the action of the group on sets of blocks and the block stabilizers. Eventually, at least for finite permutation groups, this reduces the original question to one about primitive groups. Of course, this is rarely the whole problem; generally we must then retrace the process, fitting the information back together as we reconstruct the original group, and often this is very complicated. Still, the crux of many problems in finite permutation groups lies in the study of the primitive case.