ABSTRACT
Having proved some rather deep theorems about subgroups, let us pause and take stock of what we know about specific groups. We have applied Euler’s groups directly to number theory. We have also seen groups aris ing in other contexts, such as permutations of sets of objects, invertible linear transformations of vector spaces, and symmetries of the circle (and possibly other geometric objects). Groups thereby have acquired a special significance, and deserve to be studied in their own right - the goal of such a study being to develop the tools to answer any question posed about a given group G. The most basic question is, “Does G have the same struc ture as a group we have already encountered?” or, in other words, “When are two groups the same?” To make this question precise, we consider the group structure more closely. Of course, the structure of the group is de termined by its “multiplication table,” the list of products of all pairs of elements. Here are some examples of multiplication tables of groups:
Note that the groups of the first row all have the form
where e is the neutral element and a is the other element; however the structures of (Z 4, -f) and Euler(8) differ, as seen by examining their main diagonals. Thus we see that the multiplication table provides a comprehen sive method of comparing structures of different groups (also cf. Exercise 1). But for precisely this reason, the multiplication table is far too cumbersome in most situations, and we must find a more concise method of comparing
group structures. The most direct approach is to find a correspondence of elements that will respect the algebraic structure of the groups. Even when the correspondence need not even be 1: 1, it can still transfer valuable information. We have already come across such an instance, namely the function Z -» Z m, which sends a to [a]. Let us generalize this example.
