Part II - Finite Nonabelian Groups
Published online by Cambridge University Press: 06 July 2010
Summary
The trouble is that groups behave in astonishingly subtle ways that make them psychologically rather difficult to grasp. We might say that they are adept at doing large numbers of impossible things well before breakfast.
J. H. Conway [1980, p. 170]In this part of the book we study Fourier analysis on finite nonabelian groups and we should expect them to be able to be a bit more obnoxious before breakfast than their nonabelian relatives. We will mostly deal with specific groups of 2 × 2 or 3 × 3 matrices over finite fields. Before meeting these groups for breakfast, the reader will hopefully have tasted some undergraduate algebra books such as Gallian [1990].
Brauer [1963] said: “Groups are the mathematical concept with which we describe symmetry. One of the outstanding achievements of Greek mathematics is the discovery of the 5 regular polyhedra, the Platonic solids. Each of them is closely associated with a finite group.” And we will find that these polyhedra have much to do with some of the Cayley graphs that we associate to the affine group of 2 × 2 invertible matrices over a finite field with lower row (0 1). Brauer goes on to say: “We have to confess that it took mathematicians 2000 years to achieve a mathematical formulation of the group concept. In its abstract form, it was given by Cayley, after permutation groups had been used by Lagrange, Cauchy, Abel, Galois, and others in their work on the solution of equations by radicals.”
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- Fourier Analysis on Finite Groups and Applications , pp. 237 - 239Publisher: Cambridge University PressPrint publication year: 1999