ABSTRACT
This chapter gives a broader formulation of the topics introduced with the purpose of expanding their range of applications as well as their underlying assumptions. It shortcut the notation when emphasizing the purely algebraic properties of the space F and often write, say x, y, z as points in F, thus omitting the fact that these quantities are always obtained as evaluations xs,ys,zs at points s∈V. Group actions are of practical and theoretical importance as they give a mathematical formulation to the notion of testing, or probing a quantification with certain symmetries of interest, thus allowing for assessing its invariance or regularity. The rings of the real and complex scalars, in contrast with the ring of integers, are such that the non-zero scalars form a commutative group under multiplication in which 1 is the multiplicative identity. The symmetry transformations that leave the stable configuration of a molecule physically indistinguishable are generally known as point groups.
