ABSTRACT
We introduced subgroups in the last chapter, but left many questions unanswered. For example, is there any relationship between the size of the group and the size of one of its subgroups?
In this chapter we will introduce the tool of cosets to determine many of the properties of subgroups, including what possible sizes the subgroups could be. To understand cosets, let us begin by looking at some cases where an element does not generate the group, in hopes of finding some patterns in the circle graphs. For example, consider the element 4 from the group Z10. This element does not generate the entire group, as evident from the two types of arrows in the circle graph.
