ABSTRACT
Our study of groups began in Investigation 19, where we learned about the set of symmetries of
an object. We then saw how many familiar sets, like Z, Zn, and sets of invertible square matrices, all had a structure that was similar to that of a set of symmetries-namely, the structure of a group.
One of the defining axioms of a group is that it is closed under its operation. In this investigation,
we will define and study a familiar shorthand notation for repeatedly applying a group’s operation.
Preview Activity 21.1. Use your intuition to calculate the quantities listed below. Throughout your
calculations, you will be applying the definitions that we will formally develop in this investigation.
Recall that G is the group of symmetries of a square (with the operation of composition), Z6 is the set of integers modulo 6 (with the operation of addition of congruence classes), and GL2(R) is the set of all invertible square matrices with real entries (with the operation of matrix multiplication).
