Section 4.4 Dopplegangers and the Multiverse
In the last section, we noticed that there were common behaviors among elements of groups. For example, we noticed that the order of an element or subgroup must divide the order of the group. We explored a few groups in Activity 4.3.6 that demonstrated these properties (one group was of order 6 and the other was or order 8). But what if we had another group of order 6? Would it also have the exact same behavior with one element of order 1, three elements of order 2, and two elements of order 3? Or could we have a different configuration of these elements? Also, recall that as we have stated earlier, Poincaré said,
Mathematicians do not study objects, but relations among objects; they are indifferent to the replacement of objects by others as long as relations do not change. Matter is not important, only form interests them. — Henri Poincaré.
So a natural question is, can we have two different sets and operations act in the same way? Poincaré suggested that we do not care so much about what the sets of objects are, just how that they interact with each other. In this section, we will explore this idea of behavior among different sets to see if we can infer behavior of one set of objects with an operation by looking at a different set of objects with a possibly different operation.
