Introduction

The analysis of variance shown in Table (1.13) on page 9 of Chapter 1 was a consequence of the joint action of G = {1, h, ν, o} and S5 shuffling, respectively, the rows and columns of Table (1.12). The study and data-analytic applications of these rules for shuffling the experimental labels, called group actions, are among the main objectives of the present chapter. The study of group actions and orbits is an integral part of any symmetry study, and was identified earlier in the summary of Chapter 1, on page 23, steps 3, and 4. The algebraic aspects in this and the next two chapters follow closely from Serre (1977, Part I).

Permutations

In the classification of the binary sequences in length of 4 introduced in Chapter 1, the symmetries of interest were the permutations of the four positions and the permutations of the two symbols {u, y}. These sets of permutations, together with the operation of composition of functions, share all the defining algebraic properties identified in the multiplication table of {1, ν, h, o}, characteristics of a finite group. In Section 1.7 of Chapter 1, it was shown that permutations appear in many, if not all, steps of a symmetry study. They appear as labels for the voting preference data on page 12, in matrix form as linear representations, on page 6, and as shuffling machanisms with which sets of labels could be classified and interpreted for the purpose of describing the data indexed by those labels.